JournalsjemsVol. 17, No. 6pp. 1517–1546

GCD sums from Poisson integrals and systems of dilated functions

  • Christoph Aistleitner

    Technische Universität Graz, Austria
  • István Berkes

    Technische Universität Graz, Austria
  • Kristian Seip

    University of Trondheim, Norway
GCD sums from Poisson integrals and systems of dilated functions cover
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Abstract

Upper bounds for GCD sums of the form

k,=1N(gcd(nk,n))2α(nkn)α\sum_{k,{\ell}=1}^N\frac{(\mathrm {gcd}(n_k,n_{\ell}))^{2\alpha}}{(n_k n_{\ell})^\alpha}

are established, where (nk)1kN(n_k)_{1 \leq k \leq N} is any sequence of distinct positive integers and 0<α10<\alpha \le 1; the estimate for α=1/2\alpha=1/2 solves in particular a problem of Dyer and Harman from 1986, and the estimates are optimal except possibly for α=1/2\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 Lip1/2\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 k=1Nf(nkx)\sum_{k=1}^N f(n_k x) and the a.e.\ convergence of k=1ckf(nkx)\sum_{k=1}^\infty c_k f(n_kx) when ff is 11-periodic and of bounded variation or in Lip1/2\mathrm {Lip}_{1/2}.

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. 1517–1546

DOI 10.4171/JEMS/537