Felipe A. Ramírez

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Metric Diophantine Approximation: Aspects of Recent Work

2016-09-30

Victor Beresnevich , Felipe A. Ramírez , Sanju Velani

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Cocycles over higher-rank abelian actions on quotients of semisimple Lie groups

2009-01-01

Felipe A. Ramírez

We study actions by higher-rank abelian groups on quotients of semisimpleLie groups with finite center. First, we consider actions arising from theflows of two commuting elements of the Lie algebra … We study actions by higher-rank abelian groups on quotients of semisimpleLie groups with finite center. First, we consider actions arising from theflows of two commuting elements of the Lie algebra - one nilpotent and theother semisimple. Second, we consider actions arising from two commutingunipotent flows that come from an embedded copy of$\overline{\SL(2,\RR)}^{k} \times \overline{\SL(2,\RR)}^{l}$. In bothcases we show that any smooth $\RR$-valued cocycle over the action iscohomologous to a constant cocycle via a smooth transfer function. Theseresults build on theorems of D. Mieczkowski, where the same is shown for actions on $(\SL(2,\RR) \times \SL(2,\RR))$/Γ.

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Counterexamples, covering systems, and zero-one laws for inhomogeneous approximation

2016-09-09

Felipe A. Ramírez

We develop the inhomogeneous counterpart to some key aspects of the story of the Duffin–Schaeffer Conjecture (1941). Specifically, we construct counterexamples to a number of candidates for a sans-monotonicity version … We develop the inhomogeneous counterpart to some key aspects of the story of the Duffin–Schaeffer Conjecture (1941). Specifically, we construct counterexamples to a number of candidates for a sans-monotonicity version of Szüsz’s inhomogeneous (1958) version of Khintchine’s Theorem (1924). For example, given any real sequence [Formula: see text], we build a divergent series of non-negative reals [Formula: see text] such that for any [Formula: see text], almost no real number is inhomogeneously [Formula: see text]-approximable with inhomogeneous parameter [Formula: see text]. Furthermore, given any second sequence [Formula: see text] not intersecting the rational span of [Formula: see text], and assuming a dynamical version of Erdős’ Covering Systems Conjecture (1950), we can ensure that almost every real number is inhomogeneously [Formula: see text]-approximable with any inhomogeneous parameter [Formula: see text]. Next, we prove a positive result that is near optimal in view of the limitations that our counterexamples impose. This leads to a discussion of natural analogues of the Duffin–Schaeffer Conjecture and Duffin–Schaeffer Theorem (1941) in the inhomogeneous setting. As a step toward these, we prove versions of Gallagher’s Zero-One Law (1961) for inhomogeneous approximation by reduced fractions.

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Description of some families of filiform Lie algebras

2008-01-01

Francisco Javier Echarte Reula , Juan Núñez Valdés , Felipe A. Ramírez

In this paper we describe some families of filiform Lie algebras by giving a method which allows to obtain them in any arbitrary dimension n starting from the triple (p, … In this paper we describe some families of filiform Lie algebras by giving a method which allows to obtain them in any arbitrary dimension n starting from the triple (p, q, m), where m = n and p and q are, respectively, invariants z1 and z2 of those algebras. After obtaining the general law of complex filiform Lie algebras corresponding to triples (p, q, m), some concrete examples of this method are shown

Rational approximation of affine coordinate subspaces of Euclidean space

2016-12-02

Felipe A. Ramírez , David Simmons , Fabian Süess

We show that affine coordinate subspaces of dimension at least two in Euclidean space are of Khintchine type for divergence. For affine coordinate subspaces of dimension one, we prove a … We show that affine coordinate subspaces of dimension at least two in Euclidean space are of Khintchine type for divergence. For affine coordinate subspaces of dimension one, we prove a result which depends on the dual Diophantine type of the basepoint of

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Recurrence to Shrinking Targets on Typical Self-Affine Fractals

2018-02-15

Henna Koivusalo , Felipe A. Ramírez

Abstract We explore the problem of finding the Hausdorff dimension of the set of points that recur to shrinking targets on a self-affine fractal. To be exact, we study the … Abstract We explore the problem of finding the Hausdorff dimension of the set of points that recur to shrinking targets on a self-affine fractal. To be exact, we study the dimension of a certain related symbolic recurrence set. In many cases, this set is equivalent to the recurring set on the fractal.

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Khintchine types of translated coordinate hyperplanes

2015-01-01

Felipe A. Ramírez

There has been great interest in developing a theory of “Khintchine types” for manifolds embedded in Euclidean space, and considerable progress has been made for curved manifolds. We treat the … There has been great interest in developing a theory of “Khintchine types” for manifolds embedded in Euclidean space, and considerable progress has been made for curved manifolds. We treat the case of translates of coordinate hyperplane

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Dirichlet is not just bad and singular

2022-03-17

Victor Beresnevich , Lifan Guan , Antoine Marnat +2 more

It is well known that in dimension one the set of Dirichlet improvable real numbers consists precisely of badly approximable and singular numbers. We show that in higher dimensions this … It is well known that in dimension one the set of Dirichlet improvable real numbers consists precisely of badly approximable and singular numbers. We show that in higher dimensions this is not the case by proving that there exist continuum many Dirichlet improvable vectors that are neither badly approximable nor singular. This is a consequence of a stronger statement that involves very well approximable points. In the last section we formulate the notion of intermediate Dirichlet improvable sets concerning approximations by rational planes of every intermediate dimension and show that they coincide. This naturally extends a classical theorem of Davenport & Schmidt (1969) which states that the simultaneous form of Dirichlet's theorem is improvable if and only if the dual form is improvable. Consequently, our main "continuum" result is equally valid for the corresponding intermediate Diophantine sets of badly approximable, singular and Dirichlet improvable points.

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KHINTCHINE'S THEOREM WITH RANDOM FRACTIONS

2019-12-05

Felipe A. Ramírez

We prove versions of Khintchine's theorem (1924) for approximations by rational numbers whose numerators lie in randomly chosen sets of integers, and we explore the extent to which the monotonicity … We prove versions of Khintchine's theorem (1924) for approximations by rational numbers whose numerators lie in randomly chosen sets of integers, and we explore the extent to which the monotonicity assumption can be removed. Roughly speaking, we show that if the number of available fractions for each denominator grows too fast, then the monotonicity assumption cannot be removed. There are questions in this random setting which may be seen as cognates of the Duffin–Schaeffer conjecture (1941) and are likely to be more accessible. We point out that the direct random analogue of the Duffin–Schaeffer conjecture, like the Duffin–Schaeffer conjecture itself, implies Catlin's conjecture (1976). It is not obvious whether the Duffin–Schaeffer conjecture and its random version imply one another, and it is not known whether Catlin's conjecture implies either of them. The question of whether Catlin's conjecture implies Duffin–Schaeffer conjecture has been unsettled for decades.

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Independence Inheritance and Diophantine Approximation for Systems of Linear Forms

2022-06-13

Demi Allen , Felipe A. Ramírez

Abstract The classical Khintchine–Groshev theorem is a generalization of Khintchine’s theorem on simultaneous Diophantine approximation, from approximation of points in ${\mathbb {R}}^m$ to approximation of systems of linear forms in … Abstract The classical Khintchine–Groshev theorem is a generalization of Khintchine’s theorem on simultaneous Diophantine approximation, from approximation of points in ${\mathbb {R}}^m$ to approximation of systems of linear forms in ${\mathbb {R}}^{nm}$. In this paper, we present an inhomogeneous version of the Khintchine–Groshev theorem that does not carry a monotonicity assumption when $nm&gt;2$. Our results bring the inhomogeneous theory almost in line with the homogeneous theory, where it is known by a result of Beresnevich and Velani [11] that monotonicity is not required when $nm&gt;1$. That result resolved a conjecture of Beresneich et al. [5], and our work resolves almost every case of the natural inhomogeneous generalization of that conjecture. Regarding the two cases where $nm=2$, we are able to remove monotonicity by assuming extra divergence of a measure sum, akin to a linear forms version of the Duffin–Schaeffer conjecture. When $nm=1$, it is known by work of Duffin and Schaeffer [16] that the monotonicity assumption cannot be dropped. The key new result is an independence inheritance phenomenon; the underlying idea is that the sets involved in the $((n+k)\times m)$-dimensional Khintchine–Groshev theorem ($k\geq 0$) are always $k$-levels more probabilistically independent than the sets involved the $(n\times m)$-dimensional theorem. Hence, it is shown that Khintchine’s theorem itself underpins the Khintchine–Groshev theory.

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Invariant distributions and cohomology for geodesic flows and higher cohomology of higher-rank Anosov actions

2013-05-28

Felipe A. Ramírez

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Higher cohomology for Anosov actions on certain homogeneous spaces

2013-10-01

Felipe A. Ramírez

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Khintchine's Theorem with random fractions.

2017-08-09

Felipe A. Ramírez

We prove versions of Khintchine's Theorem (1924) for approximations by rational numbers whose numerators lie in randomly chosen sets of integers, and we explore the extent to which the monotonicity … We prove versions of Khintchine's Theorem (1924) for approximations by rational numbers whose numerators lie in randomly chosen sets of integers, and we explore the extent to which the monotonicity assumption can be removed. Roughly speaking, we show that if the number of available fractions for each denominator grows too fast, then the monotonicity assumption cannot be removed. There are questions in this random setting which may be seen as cognates of the Duffin-Schaeffer Conjecture (1941), and are likely to be more accessible. We point out that the direct random analogue of the Duffin-Schaeffer Conjecture, like the Duffin-Schaeffer Conjecture itself, implies Catlin's Conjecture (1976). It is not obvious whether the Duffin-Schaeffer Conjecture and its random version imply one another, and it is not known whether Catlin's Conjecture implies either of them. The question of whether Catlin implies Duffin-Schaeffer has been unsettled for decades.

Remarks about inhomogeneous pair correlations

2021-09-06

Felipe A. Ramírez

Abstract Given an infinite subset $\mathcal{A} \subseteq\mathbb{N}$ , let A denote its smallest N elements. There is a rich and growing literature on the question of whether for typical $\alpha\in[0,1]$ … Abstract Given an infinite subset $\mathcal{A} \subseteq\mathbb{N}$ , let A denote its smallest N elements. There is a rich and growing literature on the question of whether for typical $\alpha\in[0,1]$ , the pair correlations of the set $\alpha A (\textrm{mod}\ 1)\subset [0,1]$ are asymptotically Poissonian as N increases. We define an inhomogeneous generalisation of the concept of pair correlation, and we consider the corresponding doubly metric question. Many of the results from the usual setting carry over to this new setting. Moreover, the double metricity allows us to establish some new results whose singly metric analogues are missing from the literature.

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Limit theorems for rank-one Lie groups

2014-01-29

Alexander Gorodnik , Felipe A. Ramírez

We investigate asymptotic behaviour of averaging operators for actions of simple rank-one Lie groups. It was previously known that these averaging operators converge almost everywhere, and we establish a more … We investigate asymptotic behaviour of averaging operators for actions of simple rank-one Lie groups. It was previously known that these averaging operators converge almost everywhere, and we establish a more precise asymptotic formula that describes their deviations from the limit.

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Cocycles over higher-rank abelian actions on quotients of semisimple Lie groups

2009-01-01

Felipe A. Ramírez

We study actions by higher-rank abelian groups on quotients of semisimple Lie groups with finite center. First, we consider actions arising from the flows of two commuting elements of the … We study actions by higher-rank abelian groups on quotients of semisimple Lie groups with finite center. First, we consider actions arising from the flows of two commuting elements of the Lie algebra--one nilpotent, and the other semisimple. Second, we consider actions from two commuting unipotent flows coming from two commuting embedded copies of SL(2,R). In both cases we show that any smooth real-valued cocycle over the action is cohomologous to a constant cocycle via a smooth transfer function.

Recurrence to shrinking targets on typical self-affine fractals

2014-09-26

Henna Koivusalo , Felipe A. Ramírez

We explore the problem of finding the Hausdorff dimension of the set of points that recur to shrinking targets on a self-affine fractal. To be exact, we study the dimension … We explore the problem of finding the Hausdorff dimension of the set of points that recur to shrinking targets on a self-affine fractal. To be exact, we study the dimension of a certain related symbolic recurrence set. In many cases this set is equivalent to the recurring set on the fractal.

Khintchine types of translated coordinate hyperplanes

2014-05-28

Felipe A. Ramírez

There has been great interest in developing a theory of Khintchine types for manifolds embedded in Euclidean space, and considerable progress has been made for curved manifolds. We treat the … There has been great interest in developing a theory of Khintchine types for manifolds embedded in Euclidean space, and considerable progress has been made for curved manifolds. We treat the case of translates of coordinate hyperplanes, decidedly flat manifolds. In our main results, we fix the value of one coordinate in Euclidean space and describe the set of points in the fiber over that fixed coordinate that are rationally approximable at a given rate. We identify translated coordinate hyperplanes for which there is a dichotomy as in Khintchine's Theorem: the set of rationally approximable points is null or full, according to the convergence or divergence of the series associated to the desired rate of approximation.

A characterization of bad approximability

2018-08-02

Felipe A. Ramírez

We show that badly approximable vectors are exactly those that cannot, for any inhomogeneous parameter, be inhomogeneously approximated at every monotone divergent rate.This implies in particular that Kurzweil's Theorem cannot … We show that badly approximable vectors are exactly those that cannot, for any inhomogeneous parameter, be inhomogeneously approximated at every monotone divergent rate.This implies in particular that Kurzweil's Theorem cannot be restricted to any points in the inhomogeneous part.Our results generalize to weighted approximations, and to higher irrationality exponents.

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Approximation by random fractions

2019-01-01

Laima Kaziulytė , Felipe A. Ramírez

We study approximation in the unit interval by rational numbers whose numerators are selected randomly with certain probabilities. Previous work showed that an analogue of Khintchine's Theorem holds in a … We study approximation in the unit interval by rational numbers whose numerators are selected randomly with certain probabilities. Previous work showed that an analogue of Khintchine's Theorem holds in a similar random model and raised the question of when the monotonicity assumption can be removed. Informally speaking, we show that if the probabilities in our model decay sufficiently fast as the denominator increases, then a Khintchine-like statement holds without a monotonicity assumption. Although our rate of decay of probabilities is unlikely to be optimal, it is known that such a result would not hold if the probabilities did not decay at all.

The Duffin–Schaeffer conjecture for systems of linear forms

2024-04-30

Felipe A. Ramírez

Abstract We extend the Duffin–Schaeffer conjecture to the setting of systems of linear forms in variables. That is, we establish a criterion to determine whether, for a given rate of … Abstract We extend the Duffin–Schaeffer conjecture to the setting of systems of linear forms in variables. That is, we establish a criterion to determine whether, for a given rate of approximation, almost all or almost no ‐by‐ systems of linear forms are approximable at that rate using integer vectors satisfying a natural coprimality condition. When , this is the classical 1941 Duffin–Schaeffer conjecture, which was proved in 2020 by Koukoulopoulos and Maynard. Pollington and Vaughan proved the higher dimensional version, where and , in 1990. The general statement we prove here was conjectured in 2009 by Beresnevich, Bernik, Dodson, and Velani. For approximations with no coprimality requirement, they also conjectured a generalized version of Catlin's conjecture, and in 2010, Beresnevich and Velani proved the cases of that. Catlin's classical conjecture, where , follows from the classical Duffin–Schaeffer conjecture. The remaining cases of the generalized version, where and , follow from our main result. Finally, through the mass transference principle, our main results imply their Hausdorff measure analogues, which were also conjectured by Beresnevich et al.

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Rational approximation of affine coordinate subspaces of Euclidean space

2015-10-16

Felipe A. Ramírez , David Simmons , Fabian Süess

Limit theorems for rank-one Lie groups

2012-05-22

Alexander Gorodnik , Felipe A. Ramírez

Remarks about inhomogeneous pair correlations

2020-08-05

Felipe A. Ramírez

Given an infinite subset $\mathcal A \subseteq\mathbb N$, let $A$ denote its smallest $N$ elements. There is a rich and growing literature on the question of whether for typical $\alpha\in[0,1]$, … Given an infinite subset $\mathcal A \subseteq\mathbb N$, let $A$ denote its smallest $N$ elements. There is a rich and growing literature on the question of whether for typical $\alpha\in[0,1]$, the pair correlations of the set $\alpha A \pmod 1\subset [0,1]$ are asymptotically Poissonian as $N$ increases. We define an inhomogeneous generalization of the concept of pair correlation, and we consider the corresponding doubly metric question. Many of the results from the usual setting carry over to this new setting. Moreover, the double metricity allows us to establish some new results whose singly metric analogues are missing from the literature.

A note on sequences that do not have metric Poissonian pair correlations

2020-08-05

Felipe A. Ramírez

The purpose of this note is to present a construction of sequences which do not have metric Poissonian pair correlations (MPPC) and whose additive energies grow at rates that come … The purpose of this note is to present a construction of sequences which do not have metric Poissonian pair correlations (MPPC) and whose additive energies grow at rates that come arbitrarily close to a threshold below which it is believed that all sequences have MPPC. A similar result appears in work of Lachmann and Technau and is proved using a totally different strategy. The main novelty here is the simplicity of the proof, which we arrive at by modifying a construction of Bourgain.

Higher cohomology of parabolic actions on certain homogeneous spaces

2017-03-01

Felipe A. Ramírez

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A note on sequences not having metric Poissonian pair correlations

2020-08-05

Felipe A. Ramírez

A note on sequences not having metric Poissonian pair correlations

2021-02-15

Felipe A. Ramírez

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Diophantine approximation in metric space

2022-12-02

Jonathan M. Fraser , Henna Koivusalo , Felipe A. Ramírez

Abstract Diophantine approximation is traditionally the study of how well real numbers are approximated by rationals. We propose a model for studying Diophantine approximation in an arbitrary totally bounded metric … Abstract Diophantine approximation is traditionally the study of how well real numbers are approximated by rationals. We propose a model for studying Diophantine approximation in an arbitrary totally bounded metric space where the rationals are replaced with a countable hierarchy of “well‐spread” points, which we refer to as abstract rationals . We prove various Jarník–Besicovitch type dimension bounds and investigate their sharpness.

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Independence inheritance and Diophantine approximation for systems of linear forms

2021-01-01

Demi Allen , Felipe A. Ramírez

The classical Khintchine-Groshev theorem is a generalization of Khintchine's theorem on simultaneous Diophantine approximation, from approximation of points in $\mathbb R^m$ to approximation of systems of linear forms in $\mathbb … The classical Khintchine-Groshev theorem is a generalization of Khintchine's theorem on simultaneous Diophantine approximation, from approximation of points in $\mathbb R^m$ to approximation of systems of linear forms in $\mathbb R^{nm}$. In this paper, we present an inhomogeneous version of the Khintchine-Groshev theorem which does not carry a monotonicity assumption when $nm>2$. Our results bring the inhomogeneous theory almost in line with the homogeneous theory, where it is known by a result of Beresnevich and Velani (2010) that monotonicity is not required when $nm>1$. That result resolved a conjecture of Beresnevich, Bernik, Dodson, and Velani (2009), and our work resolves almost every case of the natural inhomogeneous generalization of that conjecture. Regarding the two cases where $nm=2$, we are able to remove monotonicity by assuming extra divergence of a measure sum, akin to a linear forms version of the Duffin-Schaeffer conjecture. When $nm=1$ it is known by work of Duffin and Schaeffer (1941) that the monotonicity assumption cannot be dropped. The key new result is an independence inheritance phenomenon; the underlying idea is that the sets involved in the $((n+k)\times m)$-dimensional Khintchine-Groshev theorem ($k\geq 0$) are always $k$-levels more probabilistically independent than the sets involved the $(n\times m)$-dimensional theorem. Hence, it is shown that Khintchine's theorem itself underpins the Khintchine-Groshev theory.

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Diophantine approximation in metric space

2021-01-01

Jonathan M. Fraser , Henna Koivusalo , Felipe A. Ramírez

Diophantine approximation is traditionally the study of how well real numbers are approximated by rationals. We propose a model for studying Diophantine approximation in an arbitrary totally bounded metric space … Diophantine approximation is traditionally the study of how well real numbers are approximated by rationals. We propose a model for studying Diophantine approximation in an arbitrary totally bounded metric space where the rationals are replaced with a countable hierarchy of `well-spread' points, which we refer to as abstract rationals. We prove various Jarnik-Besicovitch type dimension bounds and investigate their sharpness.

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A note on sequences not having metric Poissonian pair correlations

2020-01-01

Felipe A. Ramírez

Rational approximation of affine coordinate subspaces of Euclidean space

2015-01-01

Felipe A. Ramírez , David Simmons , Fabian Süess

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Dirichlet is not just Bad and Singular

2020-01-01

Victor Beresnevich , Lifan Guan , Antoine Marnat +2 more

It is well known that in dimension one the set of Dirichlet improvable real numbers consists precisely of badly approximable and singular numbers. We show that in higher dimensions this … It is well known that in dimension one the set of Dirichlet improvable real numbers consists precisely of badly approximable and singular numbers. We show that in higher dimensions this is not the case by proving that there exist continuum many Dirichlet improvable vectors that are neither badly approximable nor singular. This is a consequence of a stronger statement that involves very well approximable points. In the last section we formulate the notion of intermediate Dirichlet improvable sets concerning approximations by rational planes of every intermediate dimension and show that they coincide. This naturally extends a classical theorem of Davenport and Schmidt (1969) which states that the simultaneous form of Dirichlet's theorem is improvable if and only if the dual form is improvable. Consequently, our main "continuum" result is equally valid for the corresponding intermediate Diophantine sets of badly approximable, singular and Dircihlet improvable points.

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Khintchine's Theorem with random fractions

2017-01-01

Felipe A. Ramírez

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Khintchine types of translated coordinate hyperplanes

2014-01-01

Felipe A. Ramírez

There has been great interest in developing a theory of "Khintchine types" for manifolds embedded in Euclidean space, and considerable progress has been made for curved manifolds. We treat the … There has been great interest in developing a theory of "Khintchine types" for manifolds embedded in Euclidean space, and considerable progress has been made for curved manifolds. We treat the case of translates of coordinate hyperplanes, decidedly flat manifolds. In our main results, we fix the value of one coordinate in Euclidean space and describe the set of points in the fiber over that fixed coordinate that are rationally approximable at a given rate. We identify translated coordinate hyperplanes for which there is a dichotomy as in Khintchine's Theorem: the set of rationally approximable points is null or full, according to the convergence or divergence of the series associated to the desired rate of approximation.

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Limit theorems for rank-one Lie groups

2012-01-01

Alexander Gorodnik , Felipe A. Ramírez

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Metric Diophantine Approximation: aspects of recent work

2016-01-01

Victor Beresnevich , Felipe A. Ramírez , Sanju Velani

In these notes, we begin by recalling aspects of the classical theory of metric Diophantine approximation; such as theorems of Khintchine, Jarn\'{\i}k, Duffin-Schaeffer and Gallagher. We then describe recent strengthening … In these notes, we begin by recalling aspects of the classical theory of metric Diophantine approximation; such as theorems of Khintchine, Jarn\'{\i}k, Duffin-Schaeffer and Gallagher. We then describe recent strengthening of various classical statements as well as recent developments in the area of Diophantine approximation on manifolds. The latter includes the well approximable, the badly approximable and the inhomogeneous aspects.

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Recurrence to shrinking targets on typical self-affine fractals

2014-01-01

Henna Koivusalo , Felipe A. Ramírez

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The Duffin--Schaeffer conjecture for systems of linear forms

2022-01-01

Felipe A. Ramírez

We extend the Duffin--Schaeffer conjecture to the setting of systems of $m$ linear forms in $n$ variables. That is, we establish a criterion to determine whether, for a given rate … We extend the Duffin--Schaeffer conjecture to the setting of systems of $m$ linear forms in $n$ variables. That is, we establish a criterion to determine whether, for a given rate of approximation, almost all or almost no $n$-by-$m$ systems of linear forms are approximable at that rate using integer vectors satisfying a natural coprimality condition. When $m=n=1$, this is the classical 1941 Duffin--Schaeffer conjecture, which was proved in 2020 by Koukoulopoulos and Maynard. Pollington and Vaughan proved the higher-dimensional version, where $m>1$ and $n=1$, in 1990. The general statement we prove here was conjectured in 2009 by Beresnevich, Bernik, Dodson, and Velani. For approximations with no coprimality requirement, they also conjectured a generalized version of Catlin's conjecture, and in 2010 Beresnevich and Velani proved the $m>1$ cases of that. Catlin's classical conjecture, where $m=n=1$, follows from the classical Duffin--Schaeffer conjecture. The remaining cases of the generalized version, where $m=1$ and $n>1$, follow from our main result. Finally, through the Mass Transference Principle, our main results imply their Hausdorff measure analogues, which were also conjectured by Beresnevich \emph{et al} (2009).

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Remarks about inhomogeneous pair correlations

2020-01-01

Felipe A. Ramírez

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Inhomogeneous approximation for systems of linear forms with primitivity constraints

2023-01-01

Demi Allen , Felipe A. Ramírez

We study (inhomogeneous) approximation for systems of linear forms using integer points which satisfy additional primitivity constraints. The first family of primitivity constraints we consider were introduced in 2015 by … We study (inhomogeneous) approximation for systems of linear forms using integer points which satisfy additional primitivity constraints. The first family of primitivity constraints we consider were introduced in 2015 by Dani, Laurent, and Nogueira, and are associated to partitions of the coordinate directions. Our results in this setting strengthen a theorem of Dani, Laurent, and Nogueira, and address problems posed by those same authors. The second primitivity constraints we consider are analogues of the coprimality required in the higher-dimensional Duffin--Schaeffer conjecture, posed by Sprind\v{z}uk in the 1970's and proved by Pollington and Vaughan in 1990. Here, with attention restricted to systems of linear forms in at least three variables, we prove a univariate inhomogeneous version of the Duffin--Schaeffer conjecture for systems of linear forms, the multivariate homogeneous version of which was stated by Beresnevich, Bernik, Dodson, and Velani in 2009 and recently proved by the second author.

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Metric bootstraps for limsup sets

2024-05-06

Felipe A. Ramírez

In metric Diophantine approximation, one frequently encounters the problem of showing that a limsup set has positive or full measure. Often it is a set of points in $m$-dimensional Euclidean … In metric Diophantine approximation, one frequently encounters the problem of showing that a limsup set has positive or full measure. Often it is a set of points in $m$-dimensional Euclidean space, or a set of $n$-by-$m$ systems of linear forms, satisfying some approximation condition infinitely often. The main results of this paper are bootstraps: if one can establish positive measure for such a limsup set in $m$-dimensional Euclidean space, then one can establish positive or full measure for an associated limsup set in the setting of $n$-by-$m$ systems of linear forms. Consequently, a class of $m$-dimensional results in Diophantine approximation can be bootstrapped to corresponding $n$-by-$m$-dimensional results. This leads to short proofs of existing, new, and hypothetical theorems for limsup sets that arise in the theory of systems of linear forms. We present several of these.

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The Duffin-Schaeffer conjecture with a moving target

2024-07-07

Manuel Hauke , Felipe A. Ramírez

We prove the inhomogeneous generalization of the Duffin-Schaeffer conjecture in dimension $m \geq 3$. That is, given $\mathbf{y}\in \mathbb{R}^m$ and $\psi:\mathbb{N}\to\mathbb{R}_{\geq 0}$ such that $\sum (\varphi(q)\psi(q)/q)^m = \infty$, we show … We prove the inhomogeneous generalization of the Duffin-Schaeffer conjecture in dimension $m \geq 3$. That is, given $\mathbf{y}\in \mathbb{R}^m$ and $\psi:\mathbb{N}\to\mathbb{R}_{\geq 0}$ such that $\sum (\varphi(q)\psi(q)/q)^m = \infty$, we show that for almost every $\mathbf{x} \in\mathbb{R}^m$ there are infinitely many rational vectors $\mathbf{a}/q$ such that $\vert q\mathbf{x} - \mathbf{a} - \mathbf{y}\vert<\psi(q)$ and such that each component of $\mathbf{a}$ is coprime to $q$. This is an inhomogeneous extension of a homogeneous conjecture of Sprind\v{z}uk which was itself proved in 1990 by Pollington and Vaughan. In fact, our main result generalizes Pollington-Vaughan not only to the inhomogeneous case, but also to the setting of moving targets, where the inhomogeneous parameter $\mathbf{y}$ is free to vary with $q$. In contrast, we show by an explicit construction that the (1-dimensional) inhomogeneous Duffin-Schaeffer conjecture fails to hold with a moving target, implying that any successful attack on the one-dimensional problem must use the fact that the inhomogeneous parameter is constant. We also introduce new questions regarding moving targets.

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Inhomogeneous approximation for systems of linear forms with primitivity constraints

2024-11-28

Demi Allen , Felipe A. Ramírez

Remarks about inhomogeneous pair correlations – CORRIGENDUM

2025-05-02

Felipe A. Ramírez

Remarks about inhomogeneous pair correlations – CORRIGENDUM

2025-05-02

Felipe A. Ramírez

Inhomogeneous approximation for systems of linear forms with primitivity constraints

2024-11-28

Demi Allen , Felipe A. Ramírez

The Duffin-Schaeffer conjecture with a moving target

2024-07-07

Manuel Hauke , Felipe A. Ramírez

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Metric bootstraps for limsup sets

2024-05-06

Felipe A. Ramírez

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The Duffin–Schaeffer conjecture for systems of linear forms

2024-04-30

Felipe A. Ramírez

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Inhomogeneous approximation for systems of linear forms with primitivity constraints

2023-01-01

Demi Allen , Felipe A. Ramírez

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Diophantine approximation in metric space

2022-12-02

Jonathan M. Fraser , Henna Koivusalo , Felipe A. Ramírez

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Independence Inheritance and Diophantine Approximation for Systems of Linear Forms

2022-06-13

Demi Allen , Felipe A. Ramírez

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Dirichlet is not just bad and singular

2022-03-17

Victor Beresnevich , Lifan Guan , Antoine Marnat +2 more

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The Duffin--Schaeffer conjecture for systems of linear forms

2022-01-01

Felipe A. Ramírez

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Remarks about inhomogeneous pair correlations

2021-09-06

Felipe A. Ramírez

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A note on sequences not having metric Poissonian pair correlations

2021-02-15

Felipe A. Ramírez

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Independence inheritance and Diophantine approximation for systems of linear forms

2021-01-01

Demi Allen , Felipe A. Ramírez

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Diophantine approximation in metric space

2021-01-01

Jonathan M. Fraser , Henna Koivusalo , Felipe A. Ramírez

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Remarks about inhomogeneous pair correlations

2020-08-05

Felipe A. Ramírez

A note on sequences that do not have metric Poissonian pair correlations

2020-08-05

Felipe A. Ramírez

A note on sequences not having metric Poissonian pair correlations

2020-08-05

Felipe A. Ramírez

A note on sequences not having metric Poissonian pair correlations

2020-01-01

Felipe A. Ramírez

Dirichlet is not just Bad and Singular

2020-01-01

Victor Beresnevich , Lifan Guan , Antoine Marnat +2 more

It is well known that in dimension one the set of Dirichlet improvable real numbers consists precisely of badly approximable and singular numbers. We show that in higher dimensions this … It is well known that in dimension one the set of Dirichlet improvable real numbers consists precisely of badly approximable and singular numbers. We show that in higher dimensions this is not the case by proving that there exist continuum many Dirichlet improvable vectors that are neither badly approximable nor singular. This is a consequence of a stronger statement that involves very well approximable points. In the last section we formulate the notion of intermediate Dirichlet improvable sets concerning approximations by rational planes of every intermediate dimension and show that they coincide. This naturally extends a classical theorem of Davenport and Schmidt (1969) which states that the simultaneous form of Dirichlet's theorem is improvable if and only if the dual form is improvable. Consequently, our main "continuum" result is equally valid for the corresponding intermediate Diophantine sets of badly approximable, singular and Dircihlet improvable points.

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Remarks about inhomogeneous pair correlations

2020-01-01

Felipe A. Ramírez

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KHINTCHINE'S THEOREM WITH RANDOM FRACTIONS

2019-12-05

Felipe A. Ramírez

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Approximation by random fractions

2019-01-01

Laima Kaziulytė , Felipe A. Ramírez

A characterization of bad approximability

2018-08-02

Felipe A. Ramírez

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Recurrence to Shrinking Targets on Typical Self-Affine Fractals

2018-02-15

Henna Koivusalo , Felipe A. Ramírez

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Khintchine's Theorem with random fractions.

2017-08-09

Felipe A. Ramírez

Higher cohomology of parabolic actions on certain homogeneous spaces

2017-03-01

Felipe A. Ramírez

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Khintchine's Theorem with random fractions

2017-01-01

Felipe A. Ramírez

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Rational approximation of affine coordinate subspaces of Euclidean space

2016-12-02

Felipe A. Ramírez , David Simmons , Fabian Süess

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Metric Diophantine Approximation: Aspects of Recent Work

2016-09-30

Victor Beresnevich , Felipe A. Ramírez , Sanju Velani

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Counterexamples, covering systems, and zero-one laws for inhomogeneous approximation

2016-09-09

Felipe A. Ramírez

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Metric Diophantine Approximation: aspects of recent work

2016-01-01

Victor Beresnevich , Felipe A. Ramírez , Sanju Velani

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Rational approximation of affine coordinate subspaces of Euclidean space

2015-10-16

Felipe A. Ramírez , David Simmons , Fabian Süess

Khintchine types of translated coordinate hyperplanes

2015-01-01

Felipe A. Ramírez

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Rational approximation of affine coordinate subspaces of Euclidean space

2015-01-01

Felipe A. Ramírez , David Simmons , Fabian Süess

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Recurrence to shrinking targets on typical self-affine fractals

2014-09-26

Henna Koivusalo , Felipe A. Ramírez

Khintchine types of translated coordinate hyperplanes

2014-05-28

Felipe A. Ramírez

Limit theorems for rank-one Lie groups

2014-01-29

Alexander Gorodnik , Felipe A. Ramírez

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Khintchine types of translated coordinate hyperplanes

2014-01-01

Felipe A. Ramírez

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Recurrence to shrinking targets on typical self-affine fractals

2014-01-01

Henna Koivusalo , Felipe A. Ramírez

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Higher cohomology for Anosov actions on certain homogeneous spaces

2013-10-01

Felipe A. Ramírez

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Invariant distributions and cohomology for geodesic flows and higher cohomology of higher-rank Anosov actions

2013-05-28

Felipe A. Ramírez

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Limit theorems for rank-one Lie groups

2012-05-22

Alexander Gorodnik , Felipe A. Ramírez

Limit theorems for rank-one Lie groups

2012-01-01

Alexander Gorodnik , Felipe A. Ramírez

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Cocycles over higher-rank abelian actions on quotients of semisimple Lie groups

2009-01-01

Felipe A. Ramírez

We study actions by higher-rank abelian groups on quotients of semisimple Lie groups with finite center. First, we consider actions arising from the flows of two commuting elements of the … We study actions by higher-rank abelian groups on quotients of semisimple Lie groups with finite center. First, we consider actions arising from the flows of two commuting elements of the Lie algebra--one nilpotent, and the other semisimple. Second, we consider actions from two commuting unipotent flows coming from two commuting embedded copies of SL(2,R). In both cases we show that any smooth real-valued cocycle over the action is cohomologous to a constant cocycle via a smooth transfer function.

Cocycles over higher-rank abelian actions on quotients of semisimple Lie groups

2009-01-01

Felipe A. Ramírez

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Description of some families of filiform Lie algebras

2008-01-01

Francisco Javier Echarte Reula , Juan Núñez Valdés , Felipe A. Ramírez

Coauthor	Papers Together
Henna Koivusalo	5
Demi Allen	4
Victor Beresnevich	4
Sanju Velani	4
Alexander Gorodnik	3
Fabian Süess	3
Jonathan M. Fraser	2
Antoine Marnat	2
David Simmons	2
Lifan Guan	2
Laima Kaziulytė	1
David Simmons	1
Juan Núñez Valdés	1
Francisco Javier Echarte Reula	1
Manuel Hauke	1

Khintchine’s problem in metric Diophantine approximation

1941-06-01

R. J. Duffin , A. C. Schaeffer

On the metric theory of inhomogeneous diophantine approximations

1955-01-01

Jaroslav Kurzweil

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Rational points near manifolds and metric Diophantine approximation

2011-12-07

Victor Beresnevich

This work is motivated by problems on simultaneous Diophantine approximation on manifolds, namely, establishing Khintchine and Jarník type theorems for submanifolds of R n .These problems have attracted a lot … This work is motivated by problems on simultaneous Diophantine approximation on manifolds, namely, establishing Khintchine and Jarník type theorems for submanifolds of R n .These problems have attracted a lot of interest since Kleinbock and Margulis proved a related conjecture of Alan Baker and V. G. Sprindžuk.They have been settled for planar curves but remain open in higher dimensions.In this paper, Khintchine and Jarník type divergence theorems are established for arbitrary analytic nondegenerate manifolds regardless of their dimension.The key to establishing these results is the study of the distribution of rational points near manifoldsa very attractive topic in its own right.Here, for the first time, we obtain sharp lower bounds for the number of rational points near nondegenerate manifolds in dimensions n > 2 and show that they are ubiquitous (that is uniformly distributed).

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Zur metrischen Theorie der diophantischen Approximationen

1926-12-01

A. Khintchine

Einige S�tze �ber Kettenbr�che, mit Anwendungen auf die Theorie der Diophantischen Approximationen

1924-03-01

A. Khintchine

Metric Simultaneous Diophantine Approximation

1962-01-01

Paul Gallagher

Some metrical theorems in Diophantine approximation. I

1950-04-01

J. W. S. Cassels

In this paper we are interested in statements about Diophantine approximation which are ‘almost always’ or ‘almost never’ true. Let be s non-negative functions of the positive integer n, and … In this paper we are interested in statements about Diophantine approximation which are ‘almost always’ or ‘almost never’ true. Let be s non-negative functions of the positive integer n, and let

The Duffin–Schaeffer Conjecture with extra divergence

2011-06-04

Alan Haynes , Andrew Pollington , Sanju Velani

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Additive energy and the Hausdorff dimension of the exceptional set in metric pair correlation problems

2017-10-01

Christoph Aistleitner , Gerhard Larcher , Mark Lewko

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A note on the Duffin-Schaeffer conjecture with slow divergence

2013-10-28

Christoph Aistleitner

For a non-negative function $\psi: ~ \N \mapsto \R$, let $W(\psi)$ denote the set of real numbers $x$ for which the inequality $|n x - a| < \psi(n)$ has infinitely … For a non-negative function $\psi: ~ \N \mapsto \R$, let $W(\psi)$ denote the set of real numbers $x$ for which the inequality $|n x - a| < \psi(n)$ has infinitely many coprime solutions $(a,n)$. The Duffin--Schaeffer conjecture, one of the most important unsolved problems in metric number theory, asserts that $W(\psi)$ has full measure provided {equation} \label{dsccond} \sum_{n=1}^\infty \frac{\psi(n) \varphi(n)}{n} = \infty. {equation} Recently Beresnevich, Harman, Haynes and Velani proved that $W(\psi)$ has full measure under the \emph{extra divergence} condition $$ \sum_{n=1}^\infty \frac{\psi(n) \varphi(n)}{n \exp(c (\log \log n) (\log \log \log n))} = \infty \qquad \textrm{for some $c>0$}. $$ In the present note we establish a \emph{slow divergence} counterpart of their result: $W(\psi)$ has full measure, provided\eqref{dsccond} holds and additionally there exists some $c>0$ such that $$ \sum_{n=2^{2^h}+1}^{2^{2^{h+1}}} \frac{\psi(n) \varphi(n)}{n} \leq \frac{c}{h} \qquad \textrm{for all \quad $h \geq 1$.} $$

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The Pair Correlation Function of Fractional Parts of Polynomials

1998-05-01

Ze x E v Rudnick , Peter Sarnak

First cohomology of Anosov actions of higher rank abelian groups and applications to rigidity

1994-12-01

Anatole Katok , Ralf Spatzier

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The k ‐dimensional Duffin and Schaeffer conjecture

1990-12-01

Andrew Pollington , R. C. Vaughan

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Approximation by reduced fractions

1961-10-01

Patrick Gallagher

1. Let $\{\delta(n)\}$ be a sequence of non-negative numbers.J. W. S. Cassels 1. Let $\{\delta(n)\}$ be a sequence of non-negative numbers.J. W. S. Cassels

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A Mass Transference Principle and the Duffin–Schaeffer conjecture for Hausdorff measures

2006-11-01

Victor Beresnevich , Sanju Velani

A Hausdorff measure version of the Duffin-Schaeffer conjecture in metric number theory is introduced and discussed.The general conjecture is established modulo the original conjecture.The key result is a Mass Transference … A Hausdorff measure version of the Duffin-Schaeffer conjecture in metric number theory is introduced and discussed.The general conjecture is established modulo the original conjecture.The key result is a Mass Transference Principle which allows us to transfer Lebesgue measure theoretic statements for lim sup subsets of R k to Hausdorff measure theoretic statements.In view of this, the Lebesgue theory of lim sup sets is shown to underpin the general Hausdorff theory.This is rather surprising since the latter theory is viewed to be a subtle refinement of the former.

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The Duffin–Schaeffer conjecture with extra divergence II

2012-12-03

Victor Beresnevich , Glyn Harman , Alan Haynes +1 more

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On a multi-dimensional Poissonian pair correlation concept and uniform distribution

2019-02-21

Aicke Hinrichs , Lisa Kaltenböck , Gerhard Larcher +2 more

The aim of the present article is to introduce a concept which allows to generalise the notion of Poissonian pair correlation, a second-order equidistribution property, to higher dimensions. Roughly speaking, … The aim of the present article is to introduce a concept which allows to generalise the notion of Poissonian pair correlation, a second-order equidistribution property, to higher dimensions. Roughly speaking, in the one-dimensional setting, the pair correlation statistics measures the distribution of spacings between sequence elements in the unit interval at distances of order of the mean spacing 1 / N. In the d-dimensional case, of course, the order of the mean spacing is $$1/N^{\frac{1}{d}}$$ , and—in our concept—the distance of sequence elements will be measured by the supremum-norm. Additionally, we show that, in some sense, almost all sequences satisfy this new concept and we examine the link to uniform distribution. The metrical pair correlation theory is investigated and it is proven that a class of typical low-discrepancy sequences in the high-dimensional unit cube do not have Poissonian pair correlations, which fits the existing results in the one-dimensional case.

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Flows on Homogeneous Spaces and Diophantine Approximation on Manifolds

1998-07-01

Dmitry Kleinbock , G. A. Margulis

We present a new approach to metric Diophantine approximation on manifolds based on the correspondence between approximation properties of numbers and orbit properties of certain flows on homogeneous spaces.This approach … We present a new approach to metric Diophantine approximation on manifolds based on the correspondence between approximation properties of numbers and orbit properties of certain flows on homogeneous spaces.This approach yields a new proof of a conjecture of Mahler, originally settled by V. G. Sprindžuk in 1964.We also prove several related hypotheses of Baker and Sprindžuk formulated in 1970s.The core of the proof is a theorem which generalizes and sharpens earlier results on non-divergence of unipotent flows on the space of lattices.

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GCD sums and sum-product estimates

2019-10-07

Thomas F. Bloom , Aled Walker

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Classical Metric Diophantine Approximation Revisited: The Khintchine-Groshev Theorem

2009-08-05

Victor Beresnevich , Sanju Velani

Let denote the set of ψ-approximable points in ⁠. Under the assumption that the approximating function ψ is monotonic, the classical Khintchine–Groshev theorem provides an elegant probabilistic criterion for the … Let denote the set of ψ-approximable points in ⁠. Under the assumption that the approximating function ψ is monotonic, the classical Khintchine–Groshev theorem provides an elegant probabilistic criterion for the Lebesgue measure of ⁠. The famous Duffin–Schaeffer counterexample shows that the monotonicity assumption on ψ is absolutely necessary when m = n = 1. On the other hand, it is known that monotonicity is not necessary when n ≥ 3 (Schmidt) or when n = 1 and m ≥ 2 (Gallagher). Surprisingly, when n = 2, the situation is unresolved. We deal with this remaining case and thereby remove all unnecessary conditions from the classical Khintchine–Groshev theorem. This settles a multidimensional analog of Catlin's conjecture.

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Diophantine approximation on planar curves: the convergence theory

2006-04-24

R. C. Vaughan , Sanju Velani

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THERE IS NO KHINTCHINE THRESHOLD FOR METRIC PAIR CORRELATIONS

2019-01-01

Christoph Aistleitner , Thomas Lachmann , Niclas Technau

We consider sequences of the form mod 1, where and where is a strictly increasing sequence of positive integers. If the asymptotic distribution of the pair correlations of this sequence … We consider sequences of the form mod 1, where and where is a strictly increasing sequence of positive integers. If the asymptotic distribution of the pair correlations of this sequence follows the Poissonian model for almost all in the sense of Lebesgue measure, we say that has the metric pair correlation property. Recent research has revealed a connection between the metric theory of pair correlations of such sequences, and the additive energy of truncations of . Bloom, Chow, Gafni and Walker speculated that there might be a convergence/divergence criterion which fully characterizes the metric pair correlation property in terms of the additive energy, similar to Khintchine's criterion in the metric theory of Diophantine approximation. In the present paper we give a negative answer to such speculations, by showing that such a criterion does not exist. To this end, we construct a sequence having large additive energy which, however, maintains the metric pair correlation property.

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Two problems in metric diophantine approximation, I

1976-08-01

Paul A. Catlin

Metrical theorems on fractional parts of sequences

1964-01-01

Wolfgang M. Schmidt

i) 1. Introduction.Let C be the additive group of real numbers modulo 1, and let x -* {x} be the natural mapping from the reals onto C. It is clear … i) 1. Introduction.Let C be the additive group of real numbers modulo 1, and let x -* {x} be the natural mapping from the reals onto C. It is clear what we shall mean by an interval / in C and by the length 1(1) of /.Denote the distance of the real number a to the closest integer by || a ||.The image in C of the set of reals Ç satisfying || £ -01| ^ s with given 0 and 0 < £ < 1/2 is an example of an interval

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On the metric theory of Diophantine approximation

1978-06-01

Jeffrey D. Vaaler

A conjecture of Duffin and Schaeffer states that oo is a necessary and sufficient condition that for almost all real x there are infinitely many positive integers n which satisfy … A conjecture of Duffin and Schaeffer states that oo is a necessary and sufficient condition that for almost all real x there are infinitely many positive integers n which satisfy | xa/n \ < a n n~x with (α, n) -1.The necessity of the condition is well known.We prove that the condition is also sufficient if oίn-

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On exceptional sets in the metric Poissonian pair correlations problem

2018-06-13

Thomas Lachmann , Niclas Technau

Let $$\left( a_{n}\right) _{n}$$ be a strictly increasing sequence of positive integers. Recent works uncovered a close connection between the additive energy $$E\left( A_{N}\right) $$ of the cut-offs $$A_{N}=\left\{ a_{n}\,{:}\,\,n\le … Let $$\left( a_{n}\right) _{n}$$ be a strictly increasing sequence of positive integers. Recent works uncovered a close connection between the additive energy $$E\left( A_{N}\right) $$ of the cut-offs $$A_{N}=\left\{ a_{n}\,{:}\,\,n\le N\right\} $$ , and $$\left( a_{n}\right) _{n}$$ possessing metric Poissonian pair correlations which is a metric version of a uniform distribution property of "second order". Firstly, the present article makes progress on a conjecture of Aichinger, Aistleitner, and Larcher; by sharpening a theorem of Bourgain which states that the set of $$\alpha \in \left[ 0,1\right] $$ satisfying that $$\left( \left\langle \alpha a_{n}\right\rangle \right) _{n}$$ with $$E\left( A_{N}\right) =\Omega \left( N^{3}\right) $$ does not have Poissonian pair correlations has positive Lebesgue measure. Secondly, we construct sequences with high additive energy which do not have metric Poissonian pair correlations, in a strong sense, and provide Hausdorff dimension estimates.

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Invariant distributions and time averages for horocycle flows

2003-09-15

Livio Flaminio , Giovanni Forni

There are infinitely many obstructions to the existence of smooth solutions of the cohomological equation Uu=f, where U is the vector field generating the horocycle flow on the unit tangent … There are infinitely many obstructions to the existence of smooth solutions of the cohomological equation Uu=f, where U is the vector field generating the horocycle flow on the unit tangent bundle SM of a Riemann surface M of finite area and f is a given function on SM. We study the Sobolev regularity of these obstructions, construct smooth solutions of the cohomological equation, and derive asymptotics for the ergodic averages of horocycle flows.

An Introduction to Diophantine Approximation.

1958-06-01

John C. Brixey , J. W. S. Cassels

Extremal subspaces and their submanifolds

2003-02-01

Dmitry Kleinbock

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A Khintchine-Type Theorem for Hyperplanes

2005-10-01

Anish Ghosh

The convergence case of a Khintchine-type theorem for a large class of hyperplanes is obtained. The approach to the problem is from a dynamical viewpoint, and a method due to … The convergence case of a Khintchine-type theorem for a large class of hyperplanes is obtained. The approach to the problem is from a dynamical viewpoint, and a method due to Kleinbock and Margulis is modified to prove the result.

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Diophantine approximation on planar curves and the distribution of rational points (with an Appendix by R. C. Vaughan)

2007-09-01

Victor Beresnevich , Detta Dickinson , Sanju Velani +1 more

Let C be a nondegenerate planar curve and for a real, positive decreasing function ψ let C(ψ) denote the set of simultaneously ψ-approximable points lying on C. We show that … Let C be a nondegenerate planar curve and for a real, positive decreasing function ψ let C(ψ) denote the set of simultaneously ψ-approximable points lying on C. We show that C is of Khintchine type for divergence; i.e. if a certain sum diverges then the one-dimensional Lebesgue measure on C of C(ψ) is full.We also obtain the Hausdorff measure analogue of the divergent Khintchine type result.In the case that C is a rational quadric the convergence counterparts of the divergent results are also obtained.Furthermore, for functions ψ with lower order in a critical range we determine a general, exact formula for the Hausdorff dimension of C(ψ).These results constitute the first precise and general results in the theory of simultaneous Diophantine approximation on manifolds.

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The first cohomology of parabolic actions for some higher-rank abelian groups and representation theory

2007-01-01

David Johnathan Mieczkowski

We make use of representation theory to study the first smooth almost-cohomology of some higher-rank abelian actions by parabolic operators. First, let $N$ be the upper-triangular group of $SL(2,\mathbb{C})$, $\Gamma$ … We make use of representation theory to study the first smooth almost-cohomology of some higher-rank abelian actions by parabolic operators. First, let $N$ be the upper-triangular group of $SL(2,\mathbb{C})$, $\Gamma$ any lattice and $\pi = L^2(SL(2,\mathbb{C})$/$\Gamma)$ the usual left-regular representation. We show that the first smooth almost-cohomology group $H_a^1(N, \pi)$ ≃ $H_a^1(SL(2,\mathbb{C}) , \pi)$. In addition, we show that the first smooth almost-cohomology of actions of certain higher-rank abelian groups $A$ acting by left translation on $(SL(2,\mathbb{R}) \times G)$/$\Gamma$ trivialize, where $G = SL(2,\mathbb{R})$ or $SL(2,\mathbb{C})$ and $\Gamma$ is any irreducible lattice. The abelian groups $A$ are generated by various mixtures of the diagonal and/or unipotent generators on each factor. As a consequence, for these examples we prove that the only smooth time changes for these actions are the trivial ones (up to an automorphism).

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On the Duffin-Schaeffer conjecture

2020-07-01

Dimitris Koukoulopoulos , James Maynard

Let $\psi: \mathbb{N}\to \mathbb{R}_{\ge 0}$ be an arbitrary function from the positive integers to the non-negative reals. Consider the set $\mathcal{A}$ of real numbers $\alpha$ for which there are infinitely … Let $\psi: \mathbb{N}\to \mathbb{R}_{\ge 0}$ be an arbitrary function from the positive integers to the non-negative reals. Consider the set $\mathcal{A}$ of real numbers $\alpha$ for which there are infinitely many reduced fractions $a/q$ such that $|\alpha-a/q| \le \psi(q)/q$. If $\sum_{q=1}^\infty \psi(q)\varphi(q)/q = \infty$, we show that $\mathcal{A}$ has full Lebesgue measure. This answers a question of Duffin and Schaeffer. As a corollary, we also establish a conjecture due to Catlin regarding non-reduced solutions to the inequality $|\alpha-a/q| \le \psi(q)/q)$, giving a refinement of Khinchin's Theorem.

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On the distribution of the convergents of almost all real numbers

1970-11-01

P. Erdős

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On the cohomology of certain dynamical systems

1980-01-01

Victor Guillemin , David Kazhdan

COHOMOLOGY OF DYNAMICAL SYSTEMS

1972-12-31

A N Livšic

In this paper criteria for the cohomological nullity of functions on phase spaces of various dynamical systems (U-systems, topological Markov chains, Smale systems) with coefficients in certain groups are formulated … In this paper criteria for the cohomological nullity of functions on phase spaces of various dynamical systems (U-systems, topological Markov chains, Smale systems) with coefficients in certain groups are formulated and proved, and applications of these criteria are studied. (In the simplest case of a transformation , the cohomological nullity of a real function means that .)

Counterexamples, covering systems, and zero-one laws for inhomogeneous approximation

2016-09-09

Felipe A. Ramírez

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Logarithm laws for flows on homogeneous spaces

1999-12-01

Dmitry Kleinbock , G. A. Margulis

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Higher cohomology for abelian groups of toral automorphisms II: the partially hyperbolic case, and corrigendum

2005-09-15

Anatole Katok , Svetlana Katok

In this paper we extend the results of an earlier paper, which deal with a description of the smooth untwisted cohomology for -actions by hyperbolic automorphisms of a torus, to … In this paper we extend the results of an earlier paper, which deal with a description of the smooth untwisted cohomology for -actions by hyperbolic automorphisms of a torus, to the partially hyperbolic case. Along the way we correct an error found in one of the steps in the proof for the hyperbolic case.

Metric simultaneous diophantine approximation (II)

1965-12-01

Patrick X. Gallagher

The Duffin-Schaeffer conjecture with extra divergence

2019-09-18

Christoph Aistleitner , Thomas Lachmann , Marc Munsch +2 more

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An Introduction To The Theory Of Numbers

2008-07-31

G. H. Hardy , E. M. Wright

Abstract An Introduction to the Theory of Numbers by G.H. Hardy and E. M. Wright is found on the reading list of virtually all elementary number theory courses and is … Abstract An Introduction to the Theory of Numbers by G.H. Hardy and E. M. Wright is found on the reading list of virtually all elementary number theory courses and is widely regarded as the primary and classic text in elementary number theory. Developed under the guidance of D.R. Heath-Brown this Sixth Edition of An Introduction to the Theory of Numbers has been extensively revised and updated to guide today's students through the key milestones and developments in number theory. Updates include a chapter by J.H. Silverman on one of the most important developments in number theory -- modular elliptic curves and their role in the proof of Fermat's Last Theorem -- a foreword by A. Wiles, and comprehensively updated end-of-chapter notes detailing the key developments in number theory. Suggestions for further reading are also included for the more avid reader The text retains the style and clarity of previous editions making it highly suitable for undergraduates in mathematics from the first year upwards as well as an essential reference for all number theorists.

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On Pair Correlation of Sequences

2019-03-24

Gerhard Larcher , Wolfgang Stockinger

We give a survey on the concept of Poissonian pair correlation (PPC) of sequences in the unit interval, on existing and recent results and we state a list of open … We give a survey on the concept of Poissonian pair correlation (PPC) of sequences in the unit interval, on existing and recent results and we state a list of open problems. Moreover, we present and discuss a quite recent multi-dimensional version of PPC.

Fractal Geometry: Mathematical Foundations and Applications.

1990-09-01

K. J. Falconer

Part I Foundations: mathematical background Hausdorff measure and dimension alternative definitions of dimension techniques for calculating dimensions local structure of fractals projections of fractals products of fractals intersections of fractals. … Part I Foundations: mathematical background Hausdorff measure and dimension alternative definitions of dimension techniques for calculating dimensions local structure of fractals projections of fractals products of fractals intersections of fractals. Part II Applications and examples: fractals defined by transformations examples from number theory graphs of functions examples from pure mathematics dynamical systems iteration of complex functions-Julia sets random fractals Brownian motion and Brownian surfaces multifractal measures physical applications.

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Canonical Perturbation Theory of Anosov Systems and Regularity Results for the Livsic Cohomology Equation

1986-05-01

Rafael de la Llave , J. M. Marco , Roberto Moriyón

We have studied the following problem.Given a compact symplectic manifold (M,u;) and a C°° family of C°° symplectic transformations f £ , when is it true that there exists another … We have studied the following problem.Given a compact symplectic manifold (M,u;) and a C°° family of C°° symplectic transformations f £ , when is it true that there exists another family of C°° diffeomorphisms g £ such that (*) 9eofo = f e og £ ?

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Über die metrische Theorie der diophantischen Approximation

1958-03-01

P. Szüsz

MIXING IN THE ABSENCE OF THE SHRINKING TARGET PROPERTY

2006-09-20

Bassam Fayad

Using reparametrizations of linear flows, we show that there exist area-preserving real analytic maps of the three-dimensional torus that are ‘mixing of all orders’ and do not enjoy the monotone … Using reparametrizations of linear flows, we show that there exist area-preserving real analytic maps of the three-dimensional torus that are ‘mixing of all orders’ and do not enjoy the monotone shrinking target property. Prior to that, we give a short proof of a result of Kurzweil from 1955: namely, that a translation Tα of the torus Td has the monotone shrinking target property if and only if the vector α is badly approximable (that is, of constant type). 2000 Mathematics Subject Classification 37E45, 37A25, 11J13. 2000 Mathematics Subject Classification 37E45, 37A25, 11J13.

The distribution of Farey points

1973-12-01

Harald Niederreiter

Analytic Number Theory

2009-01-01

Eberhard Freitag , Rolf Busam

Higher cohomology for Abelian groups of toral automorphisms

1995-06-01

Anatole Katok , Svetlana Katok

Abstract We give a complete description of smooth untwisted cohomology with coefficients in ℝ l for ℤ k -actions by hyperbolic automorphisms of a torus. For 1 ≤ n ≤ … Abstract We give a complete description of smooth untwisted cohomology with coefficients in ℝ l for ℤ k -actions by hyperbolic automorphisms of a torus. For 1 ≤ n ≤ k − 1 the nth cohomology trivializes, i.e. every cocycle is cohomologous to a constant cocycle via a smooth coboundary. For n = k a counterpart of the classical Livshitz Theorem holds: the cohomology class of a smooth k -cocycle is determined by periodic data.