On a Problem of Kac concerning Anisotropic Lacunary Sums
Lorenz Fruehwirth, Manuel Hauke
TL;DR
The paper addresses central limit theorems for anisotropic lacunary sums S_N = \\sum_{k\\le N} c_k f(n_k x) where the lacunary sequence (n_k) satisfies a Hadamard-type gap and the weights (c_k) obey a Lindeberg-Feller-type condition. The authors develop a martingale-based approach, partitioning indices into dense and buffer blocks and constructing a filtration to approximate lacunary sums by martingale differences, while carefully handling anisotropic weights by size-based partitioning and block mass analysis. Under Diophantine conditions that control the number of solutions to j n_k - j' n_\\ell = c, they prove central limit theorems for both general weighted sums and homogeneous-weight variants, with asymptotic variance tied to the L^2-norm of f and the weight structure. The results illuminate the delicate interplay between number-theoretic structure of (n_k) and probabilistic limit behavior, showing that minimal Diophantine control is necessary for nontrivial Gaussian limits in anisotropic lacunary settings, and they extend classical Kac-type CLTs to a broad, flexible anisotropic framework with potential implications for dynamical-systems and harmonic-analysis contexts.
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].