On a Problem of Kac concerning Anisotropic Lacunary Sums

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 $\widehat{f_k}$ decay at rate $\frac{1}{k^{1/2+\epsilon }}$, we prove central limit theorems for ${\sum }_{k\le N}{c}_{k}f(n_{k}x)$, 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].

22 pages, comments are welcome!