Almost sure asymptotics for the number variance of dilations of integer sequences

TL;DR

The paper analyzes the number variance $V(N,S,α)$ for dilations $(α x_n)$ modulo $1$, establishing Poissonian statistics $V = NS + o(NS)$ for almost all $α$ in broad ranges of $S$ under mild regularity on $(x_n)$. For polynomial sequences $x_n = p(n)$ with $ ext{deg}(p)\nobreak = obreak 1$, it proves explicit $S$-ranges where Poissonian behavior holds: up to $S \, ange \, (\,\\log N)^{-c}$ with $c=32+\\varepsilon$ for quadratics and $c=3+\\varepsilon$ for degree $ obreak \ge 3$, and an additive-energy condition $E_N = O_\\eta(N^{2+\\eta})$ yielding Poissonian behavior up to $S \,=\, N^{-\\eta-\\varepsilon}$. The paper develops a robust variance framework based on a dyadic tent-decomposition of the correlation function, connects the off-diagonal variance to representation counts and gcd-sums, and handles the quadratic and higher-degree cases with tailored bounds. It also discusses the limits of such Poissonian behavior for Kronecker sequences and for random sequences, illustrating a threshold near $S(N) \\sim 1/\\log\\log N$ in the random model. The results contribute a metric and additive-combinatorial perspective to local statistics of dilated integer sequences modulo 1, with implications for pseudorandomness and quantum-chaology models.

Abstract

Let $(x_n)_{n=1}^\infty$ be a sequence of integers. We study the number variance of dilations $(αx_n)_{n=1}^\infty$ modulo 1 in intervals of length $S$, and establish pseudorandom (Poissonian) behavior for Lebesgue-almost all $α$ throughout a large range of $S$, subject to certain regularity assumptions imposed upon $(x_n)_{n=1}^\infty$. For the important special case $x_n = p(n)$, where $p$ is a polynomial with integer coefficients of degree at least 2, we prove that the number variance is Poissonian for almost all $α$ throughout the range $0 \leq S \leq (\log N)^{-c}$, for a suitable absolute constant $c>0$. For more general sequences $(x_n)_{n=1}^\infty$, we give a criterion for Poissonian behavior for generic $α$ which is formulated in terms of the additive energy of the finite truncations $(x_n)_{n=1}^N$.

Figures (1)

• Figure 2.1: Dyadic decomposition of a tent function (left) and the way it decomposes into a sum of functions of the type $f_{v,c}$ (right). Pictured is the function $\psi_{S/2}$ for $S = \frac{15}{64} = \frac{1}{8} + \frac{1}{16} + \frac{1}{32} + \frac{1}{64}$. We have $\psi_{\frac{15}{64}} = f_{3,0} + f_{4,2} + f_{5,6} + f_{6,14}$. For this decomposition, a function $f_{v,c}$ for a certain value of $v$ is required exactly if the $v$-th digit (after the decimal point) in the binary expansion of $S$ is a "1". For such a $v$, the suitable value of $c$ can also be read off of the binary expansion of $S$, cf. Equation \ref{['eq_dyad_arise']}. Note that all functions $f_{v,c}$ in this decomposition are continuous (actually, even Lipschitz continuous with parameter 1), which affects the decay order of the Fourier coefficients and is crucially used for the variance estimate in Lemma \ref{['prop:VarianceBound']}--\ref{['prop:VarianceBound_coarse_2']}.

Theorems & Definitions (30)

• Theorem 1.1
• Proposition 1.2
• Theorem 1.3
• Proposition 1.4
• Lemma 2.1: Special case of mss
• Lemma 2.2
• proof : Proof of Lemma \ref{['prop:VarianceBound']}
• Lemma 2.3
• proof
• Lemma 2.4