GCD sums from Poisson integrals and systems of dilated functions

Aistleitner, C. and Berkes, István and Seip, K. (2015) GCD sums from Poisson integrals and systems of dilated functions. JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY, 17 (6). pp. 1517-1546. ISSN 1435-9855

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Abstract

Upper bounds for GCD sums of the form (Formula Presented) are established, where (n<inf>k</inf>)1≤k≤N is any sequence of distinct positive integers and 0 < α ≤ 1; the estimate for α = 1/2 solves in particular a problem of Dyer and Harman from 1986, and the estimates are optimal except possibly for α = 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 Lip<inf>1/2</inf>, 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 (Formula Presented)=1 f(n<inf>k</inf>x) and the a.e. convergence of (Formula Presented)=1 c<inf>k</inf>f(n<inf>k</inf>x) when f is 1-periodic and of bounded variation or in Lip<inf>1/2</inf>. © European Mathematical Society 2015.

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