A sharp threshold for arithmetic effects on the tail probabilities of lacunary sums
Christoph Aistleitner, Lorenz Fruehwirth, Joscha Prochno
TL;DR
The paper investigates how tail probabilities for lacunary sums $S_N(x)=\sum_{k=1}^N f(n_k x)$ transition from Gaussian to non-Gaussian behavior as arithmetic constraints tighten. It shows a sharp cutoff governed by the two-variable Diophantine count $L(N,a,b)$: if $L(N,a,b)=O(N/g_N)$, then tails behave like a normal distribution up to fluctuations of order $\sqrt{2\log g_N}$; this threshold is proven to be essentially optimal. The authors establish a sufficiency result via a martingale approach with block decompositions and Grama-type CLT bounds, and a matching necessity by constructing lacunary sequences that preserve the Diophantine bound yet produce tail deviations beyond the threshold. The results illuminate the precise arithmetic-analytic balance controlling tail probabilities for general lacunary sums and extend earlier CLT/LIL findings for special cases. The methods merge harmonic analysis, Diophantine estimates, and probabilistic martingale techniques to quantify when Gaussian intuition persists.
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 $\sum_{n=1}^N \cos(2πn_k x)$, considered on the probability space $[0,1]$ with Borel $σ$-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 $\cos(2 πx)$ 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.