GCD sums from Poisson integrals and systems of dilated functions

TL;DR

This paper establishes optimal bounds for GCD sums using Poisson integrals, resolving longstanding problems on the almost everywhere behavior of dilated function systems.

Contribution

It introduces a novel Poisson integral approach to bound GCD sums and applies these bounds to solve key problems in the almost everywhere convergence and growth of dilated function systems.

Findings

Proved optimal bounds for GCD sums for 0<α≤1.

Solved a 1986 problem of Dyer and Harman for α=1/2.

Established a Carleson--Hunt-type inequality for systems of dilated functions.

Abstract

Upper bounds for GCD sums of the form [\sum_{k,{\ell}=1}^N\frac{(\gcd(n_k,n_{\ell}))^{2\alpha}}{(n_k n_{\ell})^\alpha}] are proved, where $(n_k)_{1 \leq k \leq N}$ is any sequence of distinct positive integers and $0<\alpha \le 1$; the estimate for $\alpha=1/2$ solves in particular a problem of Dyer and Harman from 1986, and the estimates are optimal except possibly for $\alpha=1/2$. The method of proof is based on identifying the sum as a certain Poisson integral on a polydisc; as a byproduct, estimates for the largest eigenvalues of the associated GCD matrices are also found. The bounds for such GCD sums are used to establish a Carleson--Hunt-type inequality for systems of dilated functions of bounded variation or belonging to $\lip12$, a result that in turn settles two longstanding problems on the a.e.\ behavior of systems of dilated functions: the a.e. growth of sums of the form $\sum_{k=1}^N f(n_k x)$ and the a.e.\ convergence of $\sum_{k=1}^\infty c_k f(n_kx)$ when $f$ is 1-periodic and of bounded variation or in $\lip12$.