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.
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.
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.
Abstract We study the notion of inhomogeneous Poissonian pair correlations, proving several properties that show similarities and differences to its homogeneous counterpart. In particular, we show that sequences with inhomogeneous …
Abstract We study the notion of inhomogeneous Poissonian pair correlations, proving several properties that show similarities and differences to its homogeneous counterpart. In particular, we show that sequences with inhomogeneous Poissonian pair correlations need not be uniformly distributed, contrary to what was till recently believed.
We study the notion of inhomogeneous Poissonian pair correlations, proving several properties that show similarities and differences to its homogeneous counterpart. In particular, we show that sequences with inhomogeneous Poissonian …
We study the notion of inhomogeneous Poissonian pair correlations, proving several properties that show similarities and differences to its homogeneous counterpart. In particular, we show that sequences with inhomogeneous Poissonian pair correlations need not be uniformly distributed, contrary to what was till recently believed.
Let $(x_n)_{n=1}^{\infty}$ be a sequence on the torus $\mathbb{T}$ (normalized to length 1). A sequence $(x_n)$ is said to have Poissonian pair correlation if, for all $s>0$, $$ \lim_{N \rightarrow …
Let $(x_n)_{n=1}^{\infty}$ be a sequence on the torus $\mathbb{T}$ (normalized to length 1). A sequence $(x_n)$ is said to have Poissonian pair correlation if, for all $s>0$, $$ \lim_{N \rightarrow \infty}{ \frac{1}{N} \# \left\{ 1 \leq m \neq n \leq N: |x_m - x_n| \leq \frac{s}{N} \right\}} = 2s.$$ It is known that this implies uniform distribution of the sequence $(x_n)$. Hinrichs, Kaltenb\"ock, Larcher, Stockinger \& Ullrich extended this result to higher dimensions and showed that sequences $(x_n)$ in $[0,1]^d$ that satisfy, for all $s>0$, $$ \lim_{N \rightarrow \infty}{ \frac{1}{N} \# \left\{ 1 \leq m \neq n \leq N: \|x_m - x_n\|_{\infty} \leq \frac{s}{N} \right\}} = (2s)^d.$$ are also uniformly distributed. We prove the same result for the extension by the Euclidean norm: if a sequence $(x_n)$ in $\mathbb{T}^d$ satisfies, for all $s > 0$, $$ \lim_{N \rightarrow \infty}{ \frac{1}{N} \# \left\{ 1 \leq m \neq n \leq N: \|x_m - x_n\|_{2} \leq \frac{s}{N} \right\}} = \omega_d s^d$$ where $\omega_d$ is the volume of the unit ball, then $(x_n)$ is uniformly distributed. Our approach shows that Poissonian Pair Correlation implies an exponential sum estimate that resembles and implies the Weyl criterion.
Let $(x_n)_{n=1}^{\infty}$ be a sequence on the torus $\mathbb{T}$ (normalized to length 1). A sequence $(x_n)$ is said to have Poissonian pair correlation if, for all $s>0$, $$ \lim_{N \rightarrow …
Let $(x_n)_{n=1}^{\infty}$ be a sequence on the torus $\mathbb{T}$ (normalized to length 1). A sequence $(x_n)$ is said to have Poissonian pair correlation if, for all $s>0$,
$$ \lim_{N \rightarrow \infty}{ \frac{1}{N} \# \left\{ 1 \leq m \neq n \leq N: |x_m - x_n| \leq \frac{s}{N} \right\}} = 2s.$$ It is known that this implies uniform distribution of the sequence $(x_n)$. Hinrichs, Kaltenbock, Larcher, Stockinger \& Ullrich extended this result to higher dimensions and showed that sequences $(x_n)$ in $[0,1]^d$ that satisfy, for all $s>0$,
$$ \lim_{N \rightarrow \infty}{ \frac{1}{N} \# \left\{ 1 \leq m \neq n \leq N: \|x_m - x_n\|_{\infty} \leq \frac{s}{N} \right\}} = (2s)^d.$$ are also uniformly distributed. We prove the same result for the extension by the Euclidean norm: if a sequence $(x_n)$ in $\mathbb{T}^d$ satisfies, for all $s > 0$,
$$ \lim_{N \rightarrow \infty}{ \frac{1}{N} \# \left\{ 1 \leq m \neq n \leq N: \|x_m - x_n\|_{2} \leq \frac{s}{N} \right\}} = \omega_d s^d$$ where $\omega_d$ is the volume of the unit ball, then $(x_n)$ is uniformly distributed. Our approach shows that Poissonian Pair Correlation implies an exponential sum estimate that resembles and implies the Weyl criterion.
We say that a sequence $(x_n)_{n \in \mathbb{N}}$ in $[0,1)$ has Poissonian pair correlations if \begin{equation*} \lim_{N \to \infty} \frac{1}{N} \# \left \lbrace 1 \leq l \neq m \leq N: …
We say that a sequence $(x_n)_{n \in \mathbb{N}}$ in $[0,1)$ has Poissonian pair correlations if \begin{equation*} \lim_{N \to \infty} \frac{1}{N} \# \left \lbrace 1 \leq l \neq m \leq N: \| x_l - x_m \| \leq \frac{s}{N} \right \rbrace = 2s \end{equation*} for every $s \geq 0$. The aim of this article is twofold. First, we will establish a gap theorem which allows to deduce that a sequence $(x_n)_{n \in \mathbb{N}}$ of real numbers in $[0,1)$ having a certain weak gap structure, cannot have Poissonian pair correlations. This result covers a broad class of sequences, e.g., Kronecker sequences, the van der Corput sequence and in more general $LS$-sequences of points and digital $(t,1)$-sequences. Additionally, this theorem enables us to derive negative pair correlation properties for sequences of the form $(\lbrace a_n \alpha \rbrace)_{n \in \mathbb{N}}$, where $(a_n)_{n \in \mathbb{N}}$ is a strictly increasing sequence of integers with maximal order of additive energy, a notion that plays an important role in many fields, e.g., additive combinatorics, and is strongly connected to Poissonian pair correlation problems. These statements are not only metrical results, but hold for all possible choices of $\alpha$.
Second, in this note we study the pair correlation statistics for sequences of the form, $x_n = \lbrace b^n \alpha \rbrace, n=1, 2, 3, \ldots$, with an integer $b \geq 2$, where we choose $\alpha$ as the Stoneham number and as an infinite de Bruijn word. We will prove that both instances fail to have the Poissonian property. Throughout this article $\lbrace \cdot \rbrace$ denotes the fractional part of a real number.
We say that a sequence $(x_n)_{n \in \mathbb{N}}$ in $[0,1)$ has Poissonian pair correlations if \begin{equation*} \lim_{N \to \infty} \frac{1}{N} \# \left \lbrace 1 \leq l \neq m \leq N: …
We say that a sequence $(x_n)_{n \in \mathbb{N}}$ in $[0,1)$ has Poissonian pair correlations if \begin{equation*} \lim_{N \to \infty} \frac{1}{N} \# \left \lbrace 1 \leq l \neq m \leq N: \| x_l - x_m \| \leq \frac{s}{N} \right \rbrace = 2s \end{equation*} for every $s \geq 0$. The aim of this article is twofold. First, we will establish a gap theorem which allows to deduce that a sequence $(x_n)_{n \in \mathbb{N}}$ of real numbers in $[0,1)$ having a certain weak gap structure, cannot have Poissonian pair correlations. This result covers a broad class of sequences, e.g., Kronecker sequences, the van der Corput sequence and in more general $LS$-sequences of points and digital $(t,1)$-sequences. Additionally, this theorem enables us to derive negative pair correlation properties for sequences of the form $(\lbrace a_n \alpha \rbrace)_{n \in \mathbb{N}}$, where $(a_n)_{n \in \mathbb{N}}$ is a strictly increasing sequence of integers with maximal order of additive energy, a notion that plays an important role in many fields, e.g., additive combinatorics, and is strongly connected to Poissonian pair correlation problems. These statements are not only metrical results, but hold for all possible choices of $\alpha$. Second, in this note we study the pair correlation statistics for sequences of the form, $x_n = \lbrace b^n \alpha \rbrace, \ n=1, 2, 3, \ldots$, with an integer $b \geq 2$, where we choose $\alpha$ as the Stoneham number and as an infinite de Bruijn word. We will prove that both instances fail to have the Poissonian property. Throughout this article $\lbrace \cdot \rbrace$ denotes the fractional part of a real number.
We present the phase diagram of clusters made of two, three and four coupled Anderson impurities. All three clusters share qualitatively similar phase diagrams that include Kondo screened and unscreened …
We present the phase diagram of clusters made of two, three and four coupled Anderson impurities. All three clusters share qualitatively similar phase diagrams that include Kondo screened and unscreened regimes separated by almost critical crossover regions reflecting the proximity to barely avoided critical points. This suggests the emergence of universal paradigms that apply to clusters of arbitrary size. We discuss how these crossover regions of the impurity models might affect the approach to the Mott transition within a cluster extension of dynamical mean field theory.
We present the phase diagram of clusters made of two, three and four coupled Anderson impurities. All three clusters share qualitatively similar phase diagrams that include Kondo screened and unscreened …
We present the phase diagram of clusters made of two, three and four coupled Anderson impurities. All three clusters share qualitatively similar phase diagrams that include Kondo screened and unscreened regimes separated by almost critical crossover regions reflecting the proximity to barely avoided critical points. This suggests the emergence of universal paradigms that apply to clusters of arbitrary size. We discuss how these crossover regions of the impurity models might affect the approach to the Mott transition within a cluster extension of dynamical mean field theory.
Abstract We show that sequences of the form $\alpha n^{\theta } \pmod {1}$ with $\alpha> 0$ and $0 < \theta < \tfrac {43}{117} = \tfrac {1}{3} + 0.0341 \ldots $ …
Abstract We show that sequences of the form $\alpha n^{\theta } \pmod {1}$ with $\alpha> 0$ and $0 < \theta < \tfrac {43}{117} = \tfrac {1}{3} + 0.0341 \ldots $ have Poissonian pair correlation. This improves upon the previous result by Lutsko, Sourmelidis, and Technau, where this was established for $\alpha> 0$ and $0 < \theta < \tfrac {14}{41} = \tfrac {1}{3} + 0.0081 \ldots $. We reduce the problem of establishing Poissonian pair correlation to a counting problem using a form of amplification and the Bombieri–Iwaniec double large sieve. The counting problem is then resolved non-optimally by appealing to the bounds of Robert–Sargos and (Fouvry–Iwaniec–)Cao–Zhai. The exponent $\theta = \tfrac {2}{5}$ is the limit of our approach.
The UHF wave function may be written as a spin-contaminated \textit{pair} wave function of the APSG form, and the overlap of the alpha and beta corresponding orbitals of the UHF …
The UHF wave function may be written as a spin-contaminated \textit{pair} wave function of the APSG form, and the overlap of the alpha and beta corresponding orbitals of the UHF solution can be taken as a proxy for the strength of the correlation captured by breaking symmetry. We demonstrate this with calculations on one- and two-dimensional hydrogen clusters and make contact with the well studied Hubbard model. The UHF corresponding orbitals pair in a manner that allows a smooth evolution from doubly occupied orbitals at small distance to one in which wave function breaks symmetry, segregating the $\alpha$ and $\beta$ electrons onto distinct sublattices at large distances. By performing spin projection on these UHF solutions, we address strong correlations that are difficult to capture at intermediate distances using a single determinant. Approved for public release: LA-UR-13-22691.
The UHF wave function may be written as a spin-contaminated \textit{pair} wave function of the APSG form, and the overlap of the alpha and beta corresponding orbitals of the UHF …
The UHF wave function may be written as a spin-contaminated \textit{pair} wave function of the APSG form, and the overlap of the alpha and beta corresponding orbitals of the UHF solution can be taken as a proxy for the strength of the correlation captured by breaking symmetry. We demonstrate this with calculations on one- and two-dimensional hydrogen clusters and make contact with the well studied Hubbard model. The UHF corresponding orbitals pair in a manner that allows a smooth evolution from doubly occupied orbitals at small distance to one in which wave function breaks symmetry, segregating the $\alpha$ and $\beta$ electrons onto distinct sublattices at large distances. By performing spin projection on these UHF solutions, we address strong correlations that are difficult to capture at intermediate distances using a single determinant. Approved for public release: LA-UR-13-22691.
The UHF wave function may be written as a spin-contaminated pair wave function of the APSG form, and the overlap of the α and β corresponding orbitals of the UHF …
The UHF wave function may be written as a spin-contaminated pair wave function of the APSG form, and the overlap of the α and β corresponding orbitals of the UHF solution can be taken as a proxy for the strength of the correlation captured by breaking symmetry. We demonstrate this with calculations on one- and two-dimensional hydrogen clusters and make contact with the well studied Hubbard model. The UHF corresponding orbitals pair in a manner that allows a smooth evolution from doubly occupied orbitals at small distance to one in which wave function breaks symmetry, segregating the α and β electrons onto distinct sublattices at large distances. By performing spin projection on these UHF solutions, we address strong correlations that are difficult to capture at intermediate distances using a single determinant.
A sequence $(x_n)_{n=1}^{\infty}$ on the torus $\mathbb{T} \cong [0,1]$ is said to exhibit Poissonian pair correlation if the local gaps behave like the gaps of a Poisson random variable, i.e. …
A sequence $(x_n)_{n=1}^{\infty}$ on the torus $\mathbb{T} \cong [0,1]$ is said to exhibit Poissonian pair correlation if the local gaps behave like the gaps of a Poisson random variable, i.e. $$ \lim_{N \rightarrow \infty}{ \frac{1}{N} \# \left\{ 1 \leq m \neq n \leq N: |x_m - x_n| \leq \frac{s}{N} \right\}} = 2s \qquad \mbox{almost surely.}$$ We show that being close to Poissonian pair correlation for few values of $s$ is enough to deduce global regularity statements: if, for some~$0 < \delta < 1/2$, a set of points $\left\{x_1, \dots, x_N \right\}$ satisfies $$ \frac{1}{N}\# \left\{ 1 \leq m \neq n \leq N: |x_m - x_n| \leq \frac{s}{N} \right\} \leq (1+\delta)2s \qquad \mbox{for all} \hspace{6pt} 1 \leq s \leq (8/\delta)\sqrt{\log{N}},$$ then the discrepancy $D_N$ of the set satisfies $D_N \lesssim \delta^{1/3} + N^{-1/3}\delta^{-1/2}$. We also show that distribution properties are reflected in the global deviation from the Poissonian pair correlation $$ N^2 D_N^5 \lesssim \frac{2}{N}\int_{0}^{N/2} \left| \frac{1}{N}\# \left\{ 1 \leq m \neq n \leq N: |x_m - x_n| \leq \frac{s}{N} \right\} - 2s \right|^2 ds \lesssim N^2 D_N^2,$$ where the lower is bound is conditioned on $D_N \gtrsim N^{-1/3}$. The proofs use a connection between exponential sums, the heat kernel on $\mathbb{T}$ and spatial localization. Exponential sum estimates are obtained as a byproduct. We also describe a connection to diaphony and several open problems.
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.