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. 15171546. ISSN 14359855

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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  Hunttype 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 1periodic and of bounded variation or in Lip<inf>1/2</inf>. © European Mathematical Society 2015.
Item Type:  Article 

Uncontrolled Keywords:  Spectral norm; Polydisc; Poisson integral; GCD sums and matrices; Convergence of series of dilated functions; CarlesonHunt inequality 
Subjects:  Q Science / természettudomány > QA Mathematics / matematika 
SWORD Depositor:  MTMT SWORD 
Depositing User:  MTMT SWORD 
Date Deposited:  15 Feb 2016 21:53 
Last Modified:  15 Feb 2016 21:53 
URI:  http://real.mtak.hu/id/eprint/33507 
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