Upper bounds for GCD sums of the form
are established, where is any sequence of distinct positive integers and ; the estimate for solves in particular a problem of Dyer and Harman from 1986, and the estimates are optimal except possibly for . 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 , 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 and the a.e.\ convergence of when is -periodic and of bounded variation or in .
Cite this article
Christoph Aistleitner, István Berkes, Kristian Seip, GCD sums from Poisson integrals and systems of dilated functions. J. Eur. Math. Soc. 17 (2015), no. 6, pp. 1517–1546DOI 10.4171/JEMS/537