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Author Description

Rachel Greenfeld is an Israeli mathematician whose research spans harmonic analysis, additive combinatorics, and partial differential equations. She completed her doctorate at the Weizmann Institute of Science and later held postdoctoral positions (including at UCLA). One of her notable achievements—carried out in collaboration with Terence Tao—concerns translational tiling problems (sometimes referred to as the “periodicity conjecture”), where they established new results on how certain sets tile the integers under translations.

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Fuglede’s spectral set conjecture for convex polytopes

2017-07-14
Rachel Greenfeld , Nir Lev
Let $\Omega$ be a convex polytope in $\mathbb{R}^d$. We say that $\Omega$ is spectral if the space $L^2(\Omega)$ admits an orthogonal basis consisting of exponential functions. There is a conjecture, … Let $\Omega$ be a convex polytope in $\mathbb{R}^d$. We say that $\Omega$ is spectral if the space $L^2(\Omega)$ admits an orthogonal basis consisting of exponential functions. There is a conjecture, which goes back to Fuglede (1974), that $\Omega$ is spectral if and only if it can tile the space by translations. It is known that if $\Omega$ tiles then it is spectral, but the converse was proved only in dimension $d=2$, by Iosevich, Katz and Tao. By a result due to Kolountzakis, if a convex polytope $\Omega\subset \mathbb{R}^d$ is spectral, then it must be centrally symmetric. We prove that also all the facets of $\Omega$ are centrally symmetric. These conditions are necessary for $\Omega$ to tile by translations. We also develop an approach which allows us to prove that in dimension $d=3$, any spectral convex polytope $\Omega$ indeed tiles by translations. Thus we obtain that Fuglede's conjecture is true for convex polytopes in $\mathbb{R}^3$.
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Spectrality and tiling by cylindric domains

2016-05-02
Rachel Greenfeld , Nir Lev
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Spectrality of product domains and Fuglede’s conjecture for convex polytopes

2020-03-01
Rachel Greenfeld , Nir Lev
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A counterexample to the periodic tiling conjecture

2024-07-01
Rachel Greenfeld , Terence Tao
The periodic tiling conjecture asserts that any finite subset of a lattice $\mathbb{Z}^d$ that tiles that lattice by translations, in fact tiles periodically. In this work we disprove this conjecture … The periodic tiling conjecture asserts that any finite subset of a lattice $\mathbb{Z}^d$ that tiles that lattice by translations, in fact tiles periodically. In this work we disprove this conjecture for sufficiently large $d$, which also implies a disproof of the corresponding conjecture for Euclidean spaces $\mathbb{R}^d$. In fact, we also obtain a counterexample in a group of the form $\mathbb{Z}^2 \times G_0$ for some finite abelian $2$-group $G_0$. Our methods rely on encoding a "Sudoku puzzle" whose rows and other non-horizontal lines are constrained to lie in a certain class of "$2$-adically structured functions," in terms of certain functional equations that can be encoded in turn as a single tiling equation, and then demonstrating that solutions to this Sudoku puzzle exist, but are all non-periodic.
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Undecidable Translational Tilings with Only Two Tiles, or One Nonabelian Tile

2023-01-04
Rachel Greenfeld , Terence Tao
Abstract We construct an example of a group $$G = \mathbb {Z}^2 \times G_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>G</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>×</mml:mo><mml:msub><mml:mi>G</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math> for a finite abelian group $$G_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>G</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math> , a subset E of $$G_0$$ … Abstract We construct an example of a group $$G = \mathbb {Z}^2 \times G_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>G</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>×</mml:mo><mml:msub><mml:mi>G</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math> for a finite abelian group $$G_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>G</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math> , a subset E of $$G_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>G</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math> , and two finite subsets $$F_1,F_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>F</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math> of G , such that it is undecidable in ZFC whether $$\mathbb {Z}^2\times E$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>×</mml:mo><mml:mi>E</mml:mi></mml:mrow></mml:math> can be tiled by translations of $$F_1,F_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>F</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math> . In particular, this implies that this tiling problem is aperiodic , in the sense that (in the standard universe of ZFC) there exist translational tilings of E by the tiles $$F_1,F_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>F</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math> , but no periodic tilings. Previously, such aperiodic or undecidable translational tilings were only constructed for sets of eleven or more tiles (mostly in $$\mathbb {Z}^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:math> ). A similar construction also applies for $$G=\mathbb {Z}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>G</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mi>d</mml:mi></mml:msup></mml:mrow></mml:math> for sufficiently large d . If one allows the group $$G_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>G</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math> to be non-abelian, a variant of the construction produces an undecidable translational tiling with only one tile F . The argument proceeds by first observing that a single tiling equation is able to encode an arbitrary system of tiling equations, which in turn can encode an arbitrary system of certain functional equations once one has two or more tiles. In particular, one can use two tiles to encode tiling problems for an arbitrary number of tiles.
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Decoupling for fractal subsets of the parabola

2022-02-03
Alan Chang , Jaume de Dios Pont , Rachel Greenfeld +3 more
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Tiling, spectrality and aperiodicity of connected sets

2023-01-01
Rachel Greenfeld , Mihail N. Kolountzakis
Let $\Omega\subset \mathbb{R}^d$ be a set of finite measure. The periodic tiling conjecture suggests that if $\Omega$ tiles $\mathbb{R}^d$ by translations then it admits at least one periodic tiling. Fuglede's … Let $\Omega\subset \mathbb{R}^d$ be a set of finite measure. The periodic tiling conjecture suggests that if $\Omega$ tiles $\mathbb{R}^d$ by translations then it admits at least one periodic tiling. Fuglede's conjecture suggests that $\Omega$ admits an orthogonal basis of exponential functions if and only if it tiles $\mathbb{R}^d$ by translations. Both conjectures are known to be false in sufficiently high dimensions, with all the so-far-known counterexamples being highly disconnected. On the other hand, both conjectures are known to be true for convex sets. In this work we study these conjectures for connected sets. We show that the periodic tiling conjecture, as well as both directions of Fuglede's conjecture are false for connected sets in sufficiently high dimensions.
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The structure of translational tilings in $\mathbb{Z}^d$

2020-01-01
Rachel Greenfeld , Terence Tao
We obtain structural results on translational tilings of periodic functions in $\mathbb{Z}^d$ by finite tiles. In particular, we show that any level one tiling of a periodic set in $\mathbb{Z}^2$ … We obtain structural results on translational tilings of periodic functions in $\mathbb{Z}^d$ by finite tiles. In particular, we show that any level one tiling of a periodic set in $\mathbb{Z}^2$ must be weakly periodic (the disjoint union of sets that are individually periodic in one direction), but present a counterexample of a higher level tiling of $\mathbb{Z}^2$ that fails to be weakly periodic. We also establish a quantitative version of the two-dimensional periodic tiling conjecture which asserts that any finite tile in $\mathbb{Z}^2$ that admits a tiling, must admit a periodic tiling, by providing a polynomial bound on the period; this also gives an exponential-type bound on the computational complexity of the problem of deciding whether a given finite subset of $\mathbb{Z}^2$ tiles or not. As a byproduct of our structural theory, we also obtain an explicit formula for a universal period for all tilings of a one-dimensional tile.

An uncountable ergodic Roth theorem and applications

2022-01-01
Polona Durcik , Rachel Greenfeld , Annina Iseli +2 more
&lt;p style='text-indent:20px;'&gt;We establish an uncountable amenable ergodic Roth theorem, in which the acting group is not assumed to be countable and the space need not be separable. This generalizes a … &lt;p style='text-indent:20px;'&gt;We establish an uncountable amenable ergodic Roth theorem, in which the acting group is not assumed to be countable and the space need not be separable. This generalizes a previous result of Bergelson, McCutcheon and Zhang, and complements a result of Zorin-Kranich. We establish the following two additional results: First, a combinatorial application about triangular patterns in certain subsets of the Cartesian square of arbitrary amenable groups, extending a result of Bergelson, McCutcheon and Zhang for countable amenable groups. Second, a new uniformity aspect in the double recurrence theorem for &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ \Gamma $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-systems for uniformly amenable groups &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ \Gamma $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. Our uncountable Roth theorem is crucial in the proof of both of these results.&lt;/p&gt;
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Measurable Tilings by Abelian Group Actions

2023-03-23
Jan Grebík , Rachel Greenfeld , Václav Rozhoň +1 more
Abstract Let $X$ be a measure space with a measure-preserving action $(g,x) \mapsto g \cdot x$ of an abelian group $G$. We consider the problem of understanding the structure of … Abstract Let $X$ be a measure space with a measure-preserving action $(g,x) \mapsto g \cdot x$ of an abelian group $G$. We consider the problem of understanding the structure of measurable tilings $F \odot A = X$ of $X$ by a measurable tile $A \subset X$ translated by a finite set $F \subset G$ of shifts, thus the translates $f \cdot A$, $f \in F$ partition $X$ up to null sets. Adapting arguments from previous literature, we establish a “dilation lemma” that asserts, roughly speaking, that $F \odot A = X$ implies $F^{r} \odot A = X$ for a large family of integer dilations $r$, and use this to establish a structure theorem for such tilings analogous to that established recently by the second and fourth authors. As applications of this theorem, we completely classify those random tilings of finitely generated abelian groups that are “factors of iid”, and show that measurable tilings of a torus ${\mathbb{T}}^{d}$ can always be continuously (in fact linearly) deformed into a tiling with rational shifts, with particularly strong results in the low-dimensional cases $d=1,2$ (in particular resolving a conjecture of Conley, the first author, and Pikhurko in the $d=1$ case).
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A counterexample to the periodic tiling conjecture

2022-01-01
Rachel Greenfeld , Terence Tao
The periodic tiling conjecture asserts that any finite subset of a lattice $\mathbb{Z}^d$ which tiles that lattice by translations, in fact tiles periodically. In this work we disprove this conjecture … The periodic tiling conjecture asserts that any finite subset of a lattice $\mathbb{Z}^d$ which tiles that lattice by translations, in fact tiles periodically. In this work we disprove this conjecture for sufficiently large $d$, which also implies a disproof of the corresponding conjecture for Euclidean spaces $\mathbb{R}^d$. In fact, we also obtain a counterexample in a group of the form $\mathbb{Z}^2 \times G_0$ for some finite abelian $2$-group $G_0$. Our methods rely on encoding a "Sudoku puzzle" whose rows and other non-horizontal lines are constrained to lie in a certain class of "$2$-adically structured functions", in terms of certain functional equations that can be encoded in turn as a single tiling equation, and then demonstrating that solutions to this Sudoku puzzle exist but are all non-periodic.
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Undecidability of translational monotilings

2023-01-01
Rachel Greenfeld , Terence Tao
In the 60's, Berger famously showed that translational tilings of $\mathbb{Z}^2$ with multiple tiles are algorithmically undecidable. Recently, Bhattacharya proved the decidability of translational monotilings (tilings by translations of a … In the 60's, Berger famously showed that translational tilings of $\mathbb{Z}^2$ with multiple tiles are algorithmically undecidable. Recently, Bhattacharya proved the decidability of translational monotilings (tilings by translations of a single tile) in $\mathbb{Z}^2$. The decidability of translational monotilings in higher dimensions remained unsolved. In this paper, by combining our recently developed techniques with ideas introduced by Aanderaa and Lewis, we finally settle this problem, achieving the undecidability of translational monotilings of (periodic subsets of) virtually $\mathbb{Z}^2$ spaces, namely, spaces of the form $\mathbb{Z}^2\times G_0$, where $G_0$ is a finite Abelian group. This also implies the undecidability of translational monotilings in $\mathbb{Z}^d$, $d\geq 3$.
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Decoupling for fractal subsets of the parabola

2020-01-01
Alan Chang , Jaume de Dios Pont , Rachel Greenfeld +3 more
We consider decoupling for a fractal subset of the parabola. We reduce studying $l^{2}L^{p}$ decoupling for a fractal subset on the parabola $\{(t, t^2) : 0 \leq t \leq 1\}$ … We consider decoupling for a fractal subset of the parabola. We reduce studying $l^{2}L^{p}$ decoupling for a fractal subset on the parabola $\{(t, t^2) : 0 \leq t \leq 1\}$ to studying $l^{2}L^{p/3}$ decoupling for the projection of this subset to the interval $[0, 1]$. This generalizes the decoupling theorem of Bourgain-Demeter in the case of the parabola. Due to the sparsity and fractal like structure, this allows us to improve upon Bourgain-Demeter's decoupling theorem for the parabola. In the case when $p/3$ is an even integer we derive theoretical and computational tools to explicitly compute the associated decoupling constant for this projection to $[0, 1]$. Our ideas are inspired by the recent work on ellipsephic sets by Biggs using nested efficient congruencing.
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Additive energies on discrete cubes

2021-01-01
Jaume de Dios Pont , Rachel Greenfeld , Paata Ivanisvili +1 more
We prove that for $d\geq 0$ and $k\geq 2$, for any subset $A$ of a discrete cube $\{0,1\}^d$, the $k-$higher energy of $A$ (the number of $2k-$tuples $(a_1,a_2,\dots,a_{2k})$ in $A^{2k}$ … We prove that for $d\geq 0$ and $k\geq 2$, for any subset $A$ of a discrete cube $\{0,1\}^d$, the $k-$higher energy of $A$ (the number of $2k-$tuples $(a_1,a_2,\dots,a_{2k})$ in $A^{2k}$ with $a_1-a_2=a_3-a_4=\dots=a_{2k-1}-a_{2k}$) is at most $|A|^{\log_{2}(2^k+2)}$, and $\log_{2}(2^k+2)$ is the best possible exponent. We also show that if $d\geq 0$ and $2\leq k\leq 10$, for any subset $A$ of a discrete cube $\{0,1\}^d$, the $k-$additive energy of $A$ (the number of $2k-$tuples $(a_1,a_2,\dots,a_{2k})$ in $A^{2k}$ with $a_1+a_2+\dots+a_k=a_{k+1}+a_{k+2}+\dots+a_{2k}$) is at most $|A|^{\log_2{ \binom{2k}{k}}}$, and $\log_2{ \binom{2k}{k}}$ is the best possible exponent. We discuss the analogous problems for the sets $\{0,1,\dots,n\}^d$ for $n\geq 2$.

Spectrality of product domains and Fuglede's conjecture for convex polytopes

2018-01-07
Rachel Greenfeld , Nir Lev
A set $\Omega \subset \mathbb{R}^d$ is said to be spectral if the space $L^2(\Omega)$ has an orthogonal basis of exponential functions. It is well-known that in many respects, spectral sets … A set $\Omega \subset \mathbb{R}^d$ is said to be spectral if the space $L^2(\Omega)$ has an orthogonal basis of exponential functions. It is well-known that in many respects, spectral sets behave like sets which can tile the space by translations. This suggests a conjecture that a product set $\Omega = A \times B$ is spectral if and only if the factors $A$ and $B$ are both spectral sets. We recently proved this in the case when $A$ is an interval in dimension one. The main result of the present paper is that the conjecture is true also when $A$ is a convex polygon in two dimensions. We discuss this result in connection with the conjecture that a convex polytope $\Omega$ is spectral if and only if it can tile by translations.

Uniform syndeticity of multiple recurrence times for two commuting actions of amenable groups

2021-01-03
Polona Durcik , Rachel Greenfeld , Annina Iseli +2 more
We establish a multiple recurrence result for two commuting actions of a discrete (not necessarily countable) amenable group on a (not necessarily separable) probability algebra. Based on it, we further … We establish a multiple recurrence result for two commuting actions of a discrete (not necessarily countable) amenable group on a (not necessarily separable) probability algebra. Based on it, we further establish that for all $\varepsilon>0$, all arbitrary uniformly amenable sets $\mathcal{G}$ of discrete groups, all pairs of commuting measure-preserving actions of a group in $\mathcal{G}$ on an arbitrary probability algebra $(X,\mu)$, and all $E\in X$ with $\mu(E)\geq \varepsilon$, the set of multiple recurrence times of $E$ is syndetic in a uniform fashion over all such measure-preserving dynamical systems, where the degree of uniform syndeticity depends only on $\varepsilon$ and the function $F$ measuring the uniform amenability of $\mathcal{G}$. Novel ingredients in the proofs of our two main results are recently developed basic results in uncountable measure theory and ergodic theory, the use of conditional analysis techniques, and an ultralimit analysis of Banach densities. We also deduce from our multiple recurrence theorem that the set of triangular configurations in a dense subset of $\Gamma\times \Gamma$ is syndetic for any discrete amenable group $\Gamma$.

Undecidable translational tilings with only two tiles, or one nonabelian tile

2021-01-01
Rachel Greenfeld , Terence Tao
We construct an example of a group $G = \mathbb{Z}^2 \times G_0$ for a finite abelian group $G_0$, a subset $E$ of $G_0$, and two finite subsets $F_1,F_2$ of $G$, … We construct an example of a group $G = \mathbb{Z}^2 \times G_0$ for a finite abelian group $G_0$, a subset $E$ of $G_0$, and two finite subsets $F_1,F_2$ of $G$, such that it is undecidable in ZFC whether $\mathbb{Z}^2\times E$ can be tiled by translations of $F_1,F_2$. In particular, this implies that this tiling problem is aperiodic, in the sense that (in the standard universe of ZFC) there exist translational tilings of $E$ by the tiles $F_1,F_2$, but no periodic tilings. Previously, such aperiodic or undecidable translational tilings were only constructed for sets of eleven or more tiles (mostly in $\mathbb{Z}^2$). A similar construction also applies for $G = \mathbb{Z}^d$ for sufficiently large $d$. If one allows the group $G_0$ to be non-abelian, a variant of the construction produces an undecidable translational tiling with only one tile $F$. The argument proceeds by first observing that a single tiling equation is able to encode an arbitrary system of tiling equations, which in turn can encode an arbitrary system of certain functional equations once one has two or more tiles. In particular, one can use two tiles to encode tiling problems for an arbitrary number of tiles.

Generalized Collatz Maps with Almost Bounded Orbits.

2021-11-11
Felipe Gonçalves , Rachel Greenfeld , José Madrid
If dividing by $p$ is a mistake, multiply by $q$ and translate, and so you'll live to iterate. We show that if we define a Collatz-like map in this form … If dividing by $p$ is a mistake, multiply by $q$ and translate, and so you'll live to iterate. We show that if we define a Collatz-like map in this form then, under suitable conditions on $p$ and $q$, almost all orbits of this map attain almost bounded values. This generalizes a recent breakthrough result of Tao for the original Collatz map (i.e., $p=2$ and $q=3$). In other words, given an arbitrary growth function $N\mapsto f(N)$ we show that almost every orbit of such map with input $N$ eventually attains a value smaller than $f(N)$.

Generalized Collatz Maps with Almost Bounded Orbits

2021-01-01
Felipe Gonçalves , Rachel Greenfeld , José Madrid
If dividing by $p$ is a mistake, multiply by $q$ and translate, and so you'll live to iterate. We show that if we define a Collatz-like map in this form … If dividing by $p$ is a mistake, multiply by $q$ and translate, and so you'll live to iterate. We show that if we define a Collatz-like map in this form then, under suitable conditions on $p$ and $q$, almost all orbits of this map attain almost bounded values. This generalizes a recent breakthrough result of Tao for the original Collatz map (i.e., $p=2$ and $q=3$). In other words, given an arbitrary growth function $N\mapsto f(N)$ we show that almost every orbit of such map with input $N$ eventually attains a value smaller than $f(N)$.
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Measurable tilings by abelian group actions

2022-01-01
Jan Grebík , Rachel Greenfeld , Václav Rozhoň +1 more
Let $X$ be a measure space with a measure-preserving action $(g,x) \mapsto g \cdot x$ of an abelian group $G$. We consider the problem of understanding the structure of measurable … Let $X$ be a measure space with a measure-preserving action $(g,x) \mapsto g \cdot x$ of an abelian group $G$. We consider the problem of understanding the structure of measurable tilings $F \odot A = X$ of $X$ by a measurable tile $A \subset X$ translated by a finite set $F \subset G$ of shifts, thus the translates $f \cdot A$, $f \in F$ partition $X$ up to null sets. Adapting arguments from previous literature, we establish a "dilation lemma" that asserts, roughly speaking, that $F \odot A = X$ implies $F^r \odot A = X$ for a large family of integer dilations $r$, and use this to establish a structure theorem for such tilings analogous to that established recently by the second and fourth authors. As applications of this theorem, we completely classify those random tilings of finitely generated abelian groups that are "factors of iid", and show that measurable tilings of a torus $\mathbb{T}^d$ can always be continuously (in fact linearly) deformed into a tiling with rational shifts, with particularly strong results in the low-dimensional cases $d=1,2$ (in particular resolving a conjecture of Conley, the first author, and Pikhurko in the $d=1$ case).
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An uncountable ergodic Roth theorem and applications

2021-01-01
Polona Durcik , Rachel Greenfeld , Annina Iseli +2 more
We establish an uncountable amenable ergodic Roth theorem, in which the acting group is not assumed to be countable and the space need not be separable. This generalizes a previous … We establish an uncountable amenable ergodic Roth theorem, in which the acting group is not assumed to be countable and the space need not be separable. This generalizes a previous result of Bergelson, McCutcheon and Zhang, and complements a result of Zorin-Kranich. We establish the following two additional results: First, a combinatorial application about triangular patterns in certain subsets of the Cartesian square of arbitrary amenable groups, extending a result of Bergelson, McCutcheon and Zhang for countable amenable groups. Second, a new uniformity aspect in the double recurrence theorem for $\Gamma$-systems for arbitrary uniformly amenable groups $\Gamma$. Our uncountable Roth theorem is crucial in the proof of both of these results.
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Spectrality of product domains and Fuglede's conjecture for convex polytopes

2018-01-01
Rachel Greenfeld , Nir Lev
A set $\Omega \subset \mathbb{R}^d$ is said to be spectral if the space $L^2(\Omega)$ has an orthogonal basis of exponential functions. It is well-known that in many respects, spectral sets … A set $\Omega \subset \mathbb{R}^d$ is said to be spectral if the space $L^2(\Omega)$ has an orthogonal basis of exponential functions. It is well-known that in many respects, spectral sets "behave like" sets which can tile the space by translations. This suggests a conjecture that a product set $\Omega = A \times B$ is spectral if and only if the factors $A$ and $B$ are both spectral sets. We recently proved this in the case when $A$ is an interval in dimension one. The main result of the present paper is that the conjecture is true also when $A$ is a convex polygon in two dimensions. We discuss this result in connection with the conjecture that a convex polytope $\Omega$ is spectral if and only if it can tile by translations.
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A counterexample to the periodic tiling conjecture (announcement)

2022-01-01
Rachel Greenfeld , Terence Tao
The periodic tiling conjecture asserts that any finite subset of a lattice $\mathbb{Z^d}$ which tiles that lattice by translations, in fact tiles periodically. We announce here a disproof of this … The periodic tiling conjecture asserts that any finite subset of a lattice $\mathbb{Z^d}$ which tiles that lattice by translations, in fact tiles periodically. We announce here a disproof of this conjecture for sufficiently large $d$, which also implies a disproof of the corresponding conjecture for Euclidean spaces $\mathbb{R^d}$. In fact, we also obtain a counterexample in a group of the form $\mathbb{Z^2} \times G_0$ for some finite abelian $G_0$. Our methods rely on encoding a certain class of "$p$-adically structured functions" in terms of certain functional equations.
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On integer distance sets

2024-01-01
Rachel Greenfeld , Marina Iliopoulou , Sarah Peluse
We develop a new approach to address some classical questions concerning the size and structure of integer distance sets. Our main result is that any integer distance set in the … We develop a new approach to address some classical questions concerning the size and structure of integer distance sets. Our main result is that any integer distance set in the Euclidean plane has all but a very small number of points lying on a single line or circle. From this, we deduce a near-optimal lower bound on the diameter of any non-collinear integer distance set of size $n$ and a strong upper bound on the size of any integer distance set in $[-N,N]^2$ with no three points on a line and no four points on a circle.

Periodicity and decidability of translational tilings by rational polygonal sets

2024-08-04
Jaume de Dios Pont , Jan Grebík , Rachel Greenfeld +1 more
The periodic tiling conjecture asserts that if a region $\Sigma\subset \mathbb R^d$ tiles $\mathbb R^d$ by translations then it admits at least one fully periodic tiling. This conjecture is known … The periodic tiling conjecture asserts that if a region $\Sigma\subset \mathbb R^d$ tiles $\mathbb R^d$ by translations then it admits at least one fully periodic tiling. This conjecture is known to hold in $\mathbb R$, and recently it was disproved in sufficiently high dimensions. In this paper, we study the periodic tiling conjecture for polygonal sets: bounded open sets in $\mathbb R^2$ whose boundary is a finite union of line segments. We prove the periodic tiling conjecture for any polygonal tile whose vertices are rational. As a corollary of our argument, we also obtain the decidability of tilings by rational polygonal sets. Moreover, we prove that any translational tiling by a rational polygonal tile is weakly-periodic, i.e., can be partitioned into finitely many singly-periodic pieces.
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Periodicity and decidability of translational tilings by rational polygonal sets

2024-08-04
Jaume de Dios Pont , Jan Grebík , Rachel Greenfeld +1 more
The periodic tiling conjecture asserts that if a region $\Sigma\subset \mathbb R^d$ tiles $\mathbb R^d$ by translations then it admits at least one fully periodic tiling. This conjecture is known … The periodic tiling conjecture asserts that if a region $\Sigma\subset \mathbb R^d$ tiles $\mathbb R^d$ by translations then it admits at least one fully periodic tiling. This conjecture is known to hold in $\mathbb R$, and recently it was disproved in sufficiently high dimensions. In this paper, we study the periodic tiling conjecture for polygonal sets: bounded open sets in $\mathbb R^2$ whose boundary is a finite union of line segments. We prove the periodic tiling conjecture for any polygonal tile whose vertices are rational. As a corollary of our argument, we also obtain the decidability of tilings by rational polygonal sets. Moreover, we prove that any translational tiling by a rational polygonal tile is weakly-periodic, i.e., can be partitioned into finitely many singly-periodic pieces.
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A counterexample to the periodic tiling conjecture

2024-07-01
Rachel Greenfeld , Terence Tao
The periodic tiling conjecture asserts that any finite subset of a lattice $\mathbb{Z}^d$ that tiles that lattice by translations, in fact tiles periodically. In this work we disprove this conjecture … The periodic tiling conjecture asserts that any finite subset of a lattice $\mathbb{Z}^d$ that tiles that lattice by translations, in fact tiles periodically. In this work we disprove this conjecture for sufficiently large $d$, which also implies a disproof of the corresponding conjecture for Euclidean spaces $\mathbb{R}^d$. In fact, we also obtain a counterexample in a group of the form $\mathbb{Z}^2 \times G_0$ for some finite abelian $2$-group $G_0$. Our methods rely on encoding a "Sudoku puzzle" whose rows and other non-horizontal lines are constrained to lie in a certain class of "$2$-adically structured functions," in terms of certain functional equations that can be encoded in turn as a single tiling equation, and then demonstrating that solutions to this Sudoku puzzle exist, but are all non-periodic.
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On integer distance sets

2024-01-01
Rachel Greenfeld , Marina Iliopoulou , Sarah Peluse
We develop a new approach to address some classical questions concerning the size and structure of integer distance sets. Our main result is that any integer distance set in the … We develop a new approach to address some classical questions concerning the size and structure of integer distance sets. Our main result is that any integer distance set in the Euclidean plane has all but a very small number of points lying on a single line or circle. From this, we deduce a near-optimal lower bound on the diameter of any non-collinear integer distance set of size $n$ and a strong upper bound on the size of any integer distance set in $[-N,N]^2$ with no three points on a line and no four points on a circle.

Measurable Tilings by Abelian Group Actions

2023-03-23
Jan Grebík , Rachel Greenfeld , Václav Rozhoň +1 more
Abstract Let $X$ be a measure space with a measure-preserving action $(g,x) \mapsto g \cdot x$ of an abelian group $G$. We consider the problem of understanding the structure of … Abstract Let $X$ be a measure space with a measure-preserving action $(g,x) \mapsto g \cdot x$ of an abelian group $G$. We consider the problem of understanding the structure of measurable tilings $F \odot A = X$ of $X$ by a measurable tile $A \subset X$ translated by a finite set $F \subset G$ of shifts, thus the translates $f \cdot A$, $f \in F$ partition $X$ up to null sets. Adapting arguments from previous literature, we establish a “dilation lemma” that asserts, roughly speaking, that $F \odot A = X$ implies $F^{r} \odot A = X$ for a large family of integer dilations $r$, and use this to establish a structure theorem for such tilings analogous to that established recently by the second and fourth authors. As applications of this theorem, we completely classify those random tilings of finitely generated abelian groups that are “factors of iid”, and show that measurable tilings of a torus ${\mathbb{T}}^{d}$ can always be continuously (in fact linearly) deformed into a tiling with rational shifts, with particularly strong results in the low-dimensional cases $d=1,2$ (in particular resolving a conjecture of Conley, the first author, and Pikhurko in the $d=1$ case).
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Undecidable Translational Tilings with Only Two Tiles, or One Nonabelian Tile

2023-01-04
Rachel Greenfeld , Terence Tao
Abstract We construct an example of a group $$G = \mathbb {Z}^2 \times G_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>G</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>×</mml:mo><mml:msub><mml:mi>G</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math> for a finite abelian group $$G_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>G</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math> , a subset E of $$G_0$$ … Abstract We construct an example of a group $$G = \mathbb {Z}^2 \times G_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>G</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>×</mml:mo><mml:msub><mml:mi>G</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:mrow></mml:math> for a finite abelian group $$G_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>G</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math> , a subset E of $$G_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>G</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math> , and two finite subsets $$F_1,F_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>F</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math> of G , such that it is undecidable in ZFC whether $$\mathbb {Z}^2\times E$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msup><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup><mml:mo>×</mml:mo><mml:mi>E</mml:mi></mml:mrow></mml:math> can be tiled by translations of $$F_1,F_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>F</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math> . In particular, this implies that this tiling problem is aperiodic , in the sense that (in the standard universe of ZFC) there exist translational tilings of E by the tiles $$F_1,F_2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi>F</mml:mi><mml:mn>1</mml:mn></mml:msub><mml:mo>,</mml:mo><mml:msub><mml:mi>F</mml:mi><mml:mn>2</mml:mn></mml:msub></mml:mrow></mml:math> , but no periodic tilings. Previously, such aperiodic or undecidable translational tilings were only constructed for sets of eleven or more tiles (mostly in $$\mathbb {Z}^2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mn>2</mml:mn></mml:msup></mml:math> ). A similar construction also applies for $$G=\mathbb {Z}^d$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:mi>G</mml:mi><mml:mo>=</mml:mo><mml:msup><mml:mrow><mml:mi>Z</mml:mi></mml:mrow><mml:mi>d</mml:mi></mml:msup></mml:mrow></mml:math> for sufficiently large d . If one allows the group $$G_0$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msub><mml:mi>G</mml:mi><mml:mn>0</mml:mn></mml:msub></mml:math> to be non-abelian, a variant of the construction produces an undecidable translational tiling with only one tile F . The argument proceeds by first observing that a single tiling equation is able to encode an arbitrary system of tiling equations, which in turn can encode an arbitrary system of certain functional equations once one has two or more tiles. In particular, one can use two tiles to encode tiling problems for an arbitrary number of tiles.
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Tiling, spectrality and aperiodicity of connected sets

2023-01-01
Rachel Greenfeld , Mihail N. Kolountzakis
Let $\Omega\subset \mathbb{R}^d$ be a set of finite measure. The periodic tiling conjecture suggests that if $\Omega$ tiles $\mathbb{R}^d$ by translations then it admits at least one periodic tiling. Fuglede's … Let $\Omega\subset \mathbb{R}^d$ be a set of finite measure. The periodic tiling conjecture suggests that if $\Omega$ tiles $\mathbb{R}^d$ by translations then it admits at least one periodic tiling. Fuglede's conjecture suggests that $\Omega$ admits an orthogonal basis of exponential functions if and only if it tiles $\mathbb{R}^d$ by translations. Both conjectures are known to be false in sufficiently high dimensions, with all the so-far-known counterexamples being highly disconnected. On the other hand, both conjectures are known to be true for convex sets. In this work we study these conjectures for connected sets. We show that the periodic tiling conjecture, as well as both directions of Fuglede's conjecture are false for connected sets in sufficiently high dimensions.
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Undecidability of translational monotilings

2023-01-01
Rachel Greenfeld , Terence Tao
In the 60's, Berger famously showed that translational tilings of $\mathbb{Z}^2$ with multiple tiles are algorithmically undecidable. Recently, Bhattacharya proved the decidability of translational monotilings (tilings by translations of a … In the 60's, Berger famously showed that translational tilings of $\mathbb{Z}^2$ with multiple tiles are algorithmically undecidable. Recently, Bhattacharya proved the decidability of translational monotilings (tilings by translations of a single tile) in $\mathbb{Z}^2$. The decidability of translational monotilings in higher dimensions remained unsolved. In this paper, by combining our recently developed techniques with ideas introduced by Aanderaa and Lewis, we finally settle this problem, achieving the undecidability of translational monotilings of (periodic subsets of) virtually $\mathbb{Z}^2$ spaces, namely, spaces of the form $\mathbb{Z}^2\times G_0$, where $G_0$ is a finite Abelian group. This also implies the undecidability of translational monotilings in $\mathbb{Z}^d$, $d\geq 3$.
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Decoupling for fractal subsets of the parabola

2022-02-03
Alan Chang , Jaume de Dios Pont , Rachel Greenfeld +3 more
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An uncountable ergodic Roth theorem and applications

2022-01-01
Polona Durcik , Rachel Greenfeld , Annina Iseli +2 more
&lt;p style='text-indent:20px;'&gt;We establish an uncountable amenable ergodic Roth theorem, in which the acting group is not assumed to be countable and the space need not be separable. This generalizes a … &lt;p style='text-indent:20px;'&gt;We establish an uncountable amenable ergodic Roth theorem, in which the acting group is not assumed to be countable and the space need not be separable. This generalizes a previous result of Bergelson, McCutcheon and Zhang, and complements a result of Zorin-Kranich. We establish the following two additional results: First, a combinatorial application about triangular patterns in certain subsets of the Cartesian square of arbitrary amenable groups, extending a result of Bergelson, McCutcheon and Zhang for countable amenable groups. Second, a new uniformity aspect in the double recurrence theorem for &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ \Gamma $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;-systems for uniformly amenable groups &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ \Gamma $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. Our uncountable Roth theorem is crucial in the proof of both of these results.&lt;/p&gt;
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Measurable tilings by abelian group actions

2022-01-01
Jan Grebík , Rachel Greenfeld , Václav Rozhoň +1 more
Let $X$ be a measure space with a measure-preserving action $(g,x) \mapsto g \cdot x$ of an abelian group $G$. We consider the problem of understanding the structure of measurable … Let $X$ be a measure space with a measure-preserving action $(g,x) \mapsto g \cdot x$ of an abelian group $G$. We consider the problem of understanding the structure of measurable tilings $F \odot A = X$ of $X$ by a measurable tile $A \subset X$ translated by a finite set $F \subset G$ of shifts, thus the translates $f \cdot A$, $f \in F$ partition $X$ up to null sets. Adapting arguments from previous literature, we establish a "dilation lemma" that asserts, roughly speaking, that $F \odot A = X$ implies $F^r \odot A = X$ for a large family of integer dilations $r$, and use this to establish a structure theorem for such tilings analogous to that established recently by the second and fourth authors. As applications of this theorem, we completely classify those random tilings of finitely generated abelian groups that are "factors of iid", and show that measurable tilings of a torus $\mathbb{T}^d$ can always be continuously (in fact linearly) deformed into a tiling with rational shifts, with particularly strong results in the low-dimensional cases $d=1,2$ (in particular resolving a conjecture of Conley, the first author, and Pikhurko in the $d=1$ case).
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A counterexample to the periodic tiling conjecture (announcement)

2022-01-01
Rachel Greenfeld , Terence Tao
The periodic tiling conjecture asserts that any finite subset of a lattice $\mathbb{Z^d}$ which tiles that lattice by translations, in fact tiles periodically. We announce here a disproof of this … The periodic tiling conjecture asserts that any finite subset of a lattice $\mathbb{Z^d}$ which tiles that lattice by translations, in fact tiles periodically. We announce here a disproof of this conjecture for sufficiently large $d$, which also implies a disproof of the corresponding conjecture for Euclidean spaces $\mathbb{R^d}$. In fact, we also obtain a counterexample in a group of the form $\mathbb{Z^2} \times G_0$ for some finite abelian $G_0$. Our methods rely on encoding a certain class of "$p$-adically structured functions" in terms of certain functional equations.
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A counterexample to the periodic tiling conjecture

2022-01-01
Rachel Greenfeld , Terence Tao
The periodic tiling conjecture asserts that any finite subset of a lattice $\mathbb{Z}^d$ which tiles that lattice by translations, in fact tiles periodically. In this work we disprove this conjecture … The periodic tiling conjecture asserts that any finite subset of a lattice $\mathbb{Z}^d$ which tiles that lattice by translations, in fact tiles periodically. In this work we disprove this conjecture for sufficiently large $d$, which also implies a disproof of the corresponding conjecture for Euclidean spaces $\mathbb{R}^d$. In fact, we also obtain a counterexample in a group of the form $\mathbb{Z}^2 \times G_0$ for some finite abelian $2$-group $G_0$. Our methods rely on encoding a "Sudoku puzzle" whose rows and other non-horizontal lines are constrained to lie in a certain class of "$2$-adically structured functions", in terms of certain functional equations that can be encoded in turn as a single tiling equation, and then demonstrating that solutions to this Sudoku puzzle exist but are all non-periodic.
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Generalized Collatz Maps with Almost Bounded Orbits.

2021-11-11
Felipe Gonçalves , Rachel Greenfeld , José Madrid
If dividing by $p$ is a mistake, multiply by $q$ and translate, and so you'll live to iterate. We show that if we define a Collatz-like map in this form … If dividing by $p$ is a mistake, multiply by $q$ and translate, and so you'll live to iterate. We show that if we define a Collatz-like map in this form then, under suitable conditions on $p$ and $q$, almost all orbits of this map attain almost bounded values. This generalizes a recent breakthrough result of Tao for the original Collatz map (i.e., $p=2$ and $q=3$). In other words, given an arbitrary growth function $N\mapsto f(N)$ we show that almost every orbit of such map with input $N$ eventually attains a value smaller than $f(N)$.

Uniform syndeticity of multiple recurrence times for two commuting actions of amenable groups

2021-01-03
Polona Durcik , Rachel Greenfeld , Annina Iseli +2 more
We establish a multiple recurrence result for two commuting actions of a discrete (not necessarily countable) amenable group on a (not necessarily separable) probability algebra. Based on it, we further … We establish a multiple recurrence result for two commuting actions of a discrete (not necessarily countable) amenable group on a (not necessarily separable) probability algebra. Based on it, we further establish that for all $\varepsilon>0$, all arbitrary uniformly amenable sets $\mathcal{G}$ of discrete groups, all pairs of commuting measure-preserving actions of a group in $\mathcal{G}$ on an arbitrary probability algebra $(X,\mu)$, and all $E\in X$ with $\mu(E)\geq \varepsilon$, the set of multiple recurrence times of $E$ is syndetic in a uniform fashion over all such measure-preserving dynamical systems, where the degree of uniform syndeticity depends only on $\varepsilon$ and the function $F$ measuring the uniform amenability of $\mathcal{G}$. Novel ingredients in the proofs of our two main results are recently developed basic results in uncountable measure theory and ergodic theory, the use of conditional analysis techniques, and an ultralimit analysis of Banach densities. We also deduce from our multiple recurrence theorem that the set of triangular configurations in a dense subset of $\Gamma\times \Gamma$ is syndetic for any discrete amenable group $\Gamma$.

Undecidable translational tilings with only two tiles, or one nonabelian tile

2021-01-01
Rachel Greenfeld , Terence Tao
We construct an example of a group $G = \mathbb{Z}^2 \times G_0$ for a finite abelian group $G_0$, a subset $E$ of $G_0$, and two finite subsets $F_1,F_2$ of $G$, … We construct an example of a group $G = \mathbb{Z}^2 \times G_0$ for a finite abelian group $G_0$, a subset $E$ of $G_0$, and two finite subsets $F_1,F_2$ of $G$, such that it is undecidable in ZFC whether $\mathbb{Z}^2\times E$ can be tiled by translations of $F_1,F_2$. In particular, this implies that this tiling problem is aperiodic, in the sense that (in the standard universe of ZFC) there exist translational tilings of $E$ by the tiles $F_1,F_2$, but no periodic tilings. Previously, such aperiodic or undecidable translational tilings were only constructed for sets of eleven or more tiles (mostly in $\mathbb{Z}^2$). A similar construction also applies for $G = \mathbb{Z}^d$ for sufficiently large $d$. If one allows the group $G_0$ to be non-abelian, a variant of the construction produces an undecidable translational tiling with only one tile $F$. The argument proceeds by first observing that a single tiling equation is able to encode an arbitrary system of tiling equations, which in turn can encode an arbitrary system of certain functional equations once one has two or more tiles. In particular, one can use two tiles to encode tiling problems for an arbitrary number of tiles.

Additive energies on discrete cubes

2021-01-01
Jaume de Dios Pont , Rachel Greenfeld , Paata Ivanisvili +1 more
We prove that for $d\geq 0$ and $k\geq 2$, for any subset $A$ of a discrete cube $\{0,1\}^d$, the $k-$higher energy of $A$ (the number of $2k-$tuples $(a_1,a_2,\dots,a_{2k})$ in $A^{2k}$ … We prove that for $d\geq 0$ and $k\geq 2$, for any subset $A$ of a discrete cube $\{0,1\}^d$, the $k-$higher energy of $A$ (the number of $2k-$tuples $(a_1,a_2,\dots,a_{2k})$ in $A^{2k}$ with $a_1-a_2=a_3-a_4=\dots=a_{2k-1}-a_{2k}$) is at most $|A|^{\log_{2}(2^k+2)}$, and $\log_{2}(2^k+2)$ is the best possible exponent. We also show that if $d\geq 0$ and $2\leq k\leq 10$, for any subset $A$ of a discrete cube $\{0,1\}^d$, the $k-$additive energy of $A$ (the number of $2k-$tuples $(a_1,a_2,\dots,a_{2k})$ in $A^{2k}$ with $a_1+a_2+\dots+a_k=a_{k+1}+a_{k+2}+\dots+a_{2k}$) is at most $|A|^{\log_2{ \binom{2k}{k}}}$, and $\log_2{ \binom{2k}{k}}$ is the best possible exponent. We discuss the analogous problems for the sets $\{0,1,\dots,n\}^d$ for $n\geq 2$.

Generalized Collatz Maps with Almost Bounded Orbits

2021-01-01
Felipe Gonçalves , Rachel Greenfeld , José Madrid
If dividing by $p$ is a mistake, multiply by $q$ and translate, and so you'll live to iterate. We show that if we define a Collatz-like map in this form … If dividing by $p$ is a mistake, multiply by $q$ and translate, and so you'll live to iterate. We show that if we define a Collatz-like map in this form then, under suitable conditions on $p$ and $q$, almost all orbits of this map attain almost bounded values. This generalizes a recent breakthrough result of Tao for the original Collatz map (i.e., $p=2$ and $q=3$). In other words, given an arbitrary growth function $N\mapsto f(N)$ we show that almost every orbit of such map with input $N$ eventually attains a value smaller than $f(N)$.
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An uncountable ergodic Roth theorem and applications

2021-01-01
Polona Durcik , Rachel Greenfeld , Annina Iseli +2 more
We establish an uncountable amenable ergodic Roth theorem, in which the acting group is not assumed to be countable and the space need not be separable. This generalizes a previous … We establish an uncountable amenable ergodic Roth theorem, in which the acting group is not assumed to be countable and the space need not be separable. This generalizes a previous result of Bergelson, McCutcheon and Zhang, and complements a result of Zorin-Kranich. We establish the following two additional results: First, a combinatorial application about triangular patterns in certain subsets of the Cartesian square of arbitrary amenable groups, extending a result of Bergelson, McCutcheon and Zhang for countable amenable groups. Second, a new uniformity aspect in the double recurrence theorem for $\Gamma$-systems for arbitrary uniformly amenable groups $\Gamma$. Our uncountable Roth theorem is crucial in the proof of both of these results.
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Spectrality of product domains and Fuglede’s conjecture for convex polytopes

2020-03-01
Rachel Greenfeld , Nir Lev
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The structure of translational tilings in $\mathbb{Z}^d$

2020-01-01
Rachel Greenfeld , Terence Tao
We obtain structural results on translational tilings of periodic functions in $\mathbb{Z}^d$ by finite tiles. In particular, we show that any level one tiling of a periodic set in $\mathbb{Z}^2$ … We obtain structural results on translational tilings of periodic functions in $\mathbb{Z}^d$ by finite tiles. In particular, we show that any level one tiling of a periodic set in $\mathbb{Z}^2$ must be weakly periodic (the disjoint union of sets that are individually periodic in one direction), but present a counterexample of a higher level tiling of $\mathbb{Z}^2$ that fails to be weakly periodic. We also establish a quantitative version of the two-dimensional periodic tiling conjecture which asserts that any finite tile in $\mathbb{Z}^2$ that admits a tiling, must admit a periodic tiling, by providing a polynomial bound on the period; this also gives an exponential-type bound on the computational complexity of the problem of deciding whether a given finite subset of $\mathbb{Z}^2$ tiles or not. As a byproduct of our structural theory, we also obtain an explicit formula for a universal period for all tilings of a one-dimensional tile.

Decoupling for fractal subsets of the parabola

2020-01-01
Alan Chang , Jaume de Dios Pont , Rachel Greenfeld +3 more
We consider decoupling for a fractal subset of the parabola. We reduce studying $l^{2}L^{p}$ decoupling for a fractal subset on the parabola $\{(t, t^2) : 0 \leq t \leq 1\}$ … We consider decoupling for a fractal subset of the parabola. We reduce studying $l^{2}L^{p}$ decoupling for a fractal subset on the parabola $\{(t, t^2) : 0 \leq t \leq 1\}$ to studying $l^{2}L^{p/3}$ decoupling for the projection of this subset to the interval $[0, 1]$. This generalizes the decoupling theorem of Bourgain-Demeter in the case of the parabola. Due to the sparsity and fractal like structure, this allows us to improve upon Bourgain-Demeter's decoupling theorem for the parabola. In the case when $p/3$ is an even integer we derive theoretical and computational tools to explicitly compute the associated decoupling constant for this projection to $[0, 1]$. Our ideas are inspired by the recent work on ellipsephic sets by Biggs using nested efficient congruencing.
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Spectrality of product domains and Fuglede's conjecture for convex polytopes

2018-01-07
Rachel Greenfeld , Nir Lev
A set $\Omega \subset \mathbb{R}^d$ is said to be spectral if the space $L^2(\Omega)$ has an orthogonal basis of exponential functions. It is well-known that in many respects, spectral sets … A set $\Omega \subset \mathbb{R}^d$ is said to be spectral if the space $L^2(\Omega)$ has an orthogonal basis of exponential functions. It is well-known that in many respects, spectral sets behave like sets which can tile the space by translations. This suggests a conjecture that a product set $\Omega = A \times B$ is spectral if and only if the factors $A$ and $B$ are both spectral sets. We recently proved this in the case when $A$ is an interval in dimension one. The main result of the present paper is that the conjecture is true also when $A$ is a convex polygon in two dimensions. We discuss this result in connection with the conjecture that a convex polytope $\Omega$ is spectral if and only if it can tile by translations.

Spectrality of product domains and Fuglede's conjecture for convex polytopes

2018-01-01
Rachel Greenfeld , Nir Lev
A set $\Omega \subset \mathbb{R}^d$ is said to be spectral if the space $L^2(\Omega)$ has an orthogonal basis of exponential functions. It is well-known that in many respects, spectral sets … A set $\Omega \subset \mathbb{R}^d$ is said to be spectral if the space $L^2(\Omega)$ has an orthogonal basis of exponential functions. It is well-known that in many respects, spectral sets "behave like" sets which can tile the space by translations. This suggests a conjecture that a product set $\Omega = A \times B$ is spectral if and only if the factors $A$ and $B$ are both spectral sets. We recently proved this in the case when $A$ is an interval in dimension one. The main result of the present paper is that the conjecture is true also when $A$ is a convex polygon in two dimensions. We discuss this result in connection with the conjecture that a convex polytope $\Omega$ is spectral if and only if it can tile by translations.
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Fuglede’s spectral set conjecture for convex polytopes

2017-07-14
Rachel Greenfeld , Nir Lev
Let $\Omega$ be a convex polytope in $\mathbb{R}^d$. We say that $\Omega$ is spectral if the space $L^2(\Omega)$ admits an orthogonal basis consisting of exponential functions. There is a conjecture, … Let $\Omega$ be a convex polytope in $\mathbb{R}^d$. We say that $\Omega$ is spectral if the space $L^2(\Omega)$ admits an orthogonal basis consisting of exponential functions. There is a conjecture, which goes back to Fuglede (1974), that $\Omega$ is spectral if and only if it can tile the space by translations. It is known that if $\Omega$ tiles then it is spectral, but the converse was proved only in dimension $d=2$, by Iosevich, Katz and Tao. By a result due to Kolountzakis, if a convex polytope $\Omega\subset \mathbb{R}^d$ is spectral, then it must be centrally symmetric. We prove that also all the facets of $\Omega$ are centrally symmetric. These conditions are necessary for $\Omega$ to tile by translations. We also develop an approach which allows us to prove that in dimension $d=3$, any spectral convex polytope $\Omega$ indeed tiles by translations. Thus we obtain that Fuglede's conjecture is true for convex polytopes in $\mathbb{R}^3$.
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Spectrality and tiling by cylindric domains

2016-05-02
Rachel Greenfeld , Nir Lev
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Coauthor Papers Together
José Madrid 9
Terence Tao 9
Nir Lev 5
Asgar Jamneshan 5
Jaume de Dios Pont 4
Polona Durcik 3
Jan Grebík 3
Annina Iseli 3
Zane Kun Li 2
Felipe Gonçalves 2
Alan Chang 2
Václav Rozhoň 2
Marina Iliopoulou 1
Mihail N. Kolountzakis 1
Sarah Peluse 1
Paata Ivanisvili 1