2012/10/02 by Christoph Aistleitner, Aistleitner, Christoph, Istvan Berkes +3 · 1 citation
Mathematics · #Classical Analysis and ODEs (math.CA) #Complex Variables (math.CV) #FOS: Mathematics #Functional Analysis (math.FA) #Number Theory (math.NT) #math.CA #math.CV #math.FA #math.NT
paper · pdf · doi:10.48550/arxiv.1210.0741
arxiv created 2013/11/11 · arxiv updated 2013/11/12
Upper bounds for GCD sums of the form [∑_k,ℓ=1N\frac(gcd(nk,nℓ))2α(nk nℓ)α] are proved, where (nk)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 \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 ∑k=1N f(nk x) and the a.e. convergence of ∑k=1^∞ ck f(nkx) when f is 1-periodic and of bounded variation or in \lip12.