GCD sums from Poisson integrals and systems of dilated functions

Abstract

Upper bounds for GCD sums of the form

are established, where $(n_k{)}_{1\le k\le N}$ 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 ${\mathrm{Lip}}_{1/2}$, 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 ${\sum }_{k=1}^{\infty }{c}_{k}f(n_{k}x)$ when $f$ is $1$-periodic and of bounded variation or in ${\mathrm{Lip}}_{1/2}$.