A sharp threshold for arithmetic effects on the tail probabilities of lacunary sums

TL;DR

This paper establishes a precise Diophantine criterion determining when lacunary sums of functions exhibit Gaussian tail behavior or deviate, revealing a sharp threshold based on the solutions of certain Diophantine equations.

Contribution

The authors identify a sharp Diophantine threshold that dictates the transition from Gaussian to erratic tail behavior in lacunary sums, extending previous results on CLT and LIL.

Findings

Threshold criterion based on solutions to Diophantine equations

Gaussian tail behavior holds below the threshold

Behavior can deviate beyond the threshold

Abstract

A classical observation in analysis asserts that lacunary systems of dilated functions show many properties which are also typical for systems of independent random variables. For example, if $(n_k)_{k \ge 1}$ is a sequence of integers satisfying the Hadamard gap condition $n_{k+1}/n_k\ge q > 1,~k \ge 1$, then the normalized sums $\sum_{n=1}^N \cos(2\pi n_k x)$, considered on the probability space $[0,1]$ with Borel $\sigma$-field and Lebesgue measure, satisfy the central limit theorem (CLT) and the law of the iterated logarithm (LIL). Remarkably, the situation becomes much more deliacate when the trigonometric function $\cos(2 \pi x)$ is replaced by a more general 1-periodic function $f$, and fine arithmetic properties of the sequence $(n_k)_{k \ge 1}$ come into play. The most relevant arithmetic property can be phrased in terms of the number of solutions of certain 2-variable Diophantine equations. Recently, the authors proved that the validity of the LIL requires a strictly stronger Diophantine criterion than the CLT. In the present paper we show that this is only a special case of a wide-ranging general principle: there is a sharp cutoff, which can be expressed in form of a Diophantine criterion on the sequence $(n_k)_{k \ge 1}$, at which the tail probabilities of $\sum_{k=1}^N f(n_k x)$ change from Gaussian to potentially erratic behavior. More precisely, let $L(N,a,b,c)$ be the number of solutions $(k,\ell)$ of the equation $a n_k - b n_\ell= c$, where $1\leq k,\ell \leq N$. Roughly speaking, we prove: if $L(N,a,b,c) \le N / g_N$ for some $g_N$, then $\mathbb{P} \left[\sum_{k=1}^N f(n_k x) > t \|f\|_2 \sqrt{N} \right]$ is asymptotically is accordance with standard normal behavior for all $t$ up to $\sqrt{2 \log g_N}$. We also show that this criterion is optimal in the sense that under the same premises, the conclusion can fail to be true for values of $t$ beyond this threshold.