On a Problem of Kac concerning Anisotropic Lacunary Sums

TL;DR

This paper establishes central limit theorems for lacunary sums with weighted functions under Diophantine conditions, addressing a question posed by Kac about anisotropic lacunary sums and their probabilistic behavior.

Contribution

It proves CLTs for weighted lacunary sums with functions having specific Fourier decay, under necessary Diophantine conditions, extending Kac's classical problem.

Findings

Proves CLTs for lacunary sums with weights and Fourier decay.

Identifies Diophantine conditions as necessary for the results.

Addresses a historical question by Kac from 1946.

Abstract

Given a lacunary sequence $(n_k)_{k \in \mathbb{N}}$, arbitrary positive weights $(c_k)_{k \in \mathbb{N}}$ that satisfy a Lindeberg-Feller condition, and a function $f: \mathbb{T} \to \mathbb{R}$ whose Fourier coefficients $\hat{f_k}$ decay at rate $\frac{1}{k^{1/2 + \varepsilon}}$, we prove central limit theorems for $\sum_{k \leq N}c_kf(n_kx)$, provided $(n_k)_{k \in \mathbb{N}}$ satisfies a Diophantine condition that is necessary in general. This addresses a question raised by M. Kac [Ann. of Math., 1946].