GCD sums from Poisson integrals and systems of dilated functions

Aistleitner, Christoph, Berkes, István, and Seip, Kristian. "GCD sums from Poisson integrals and systems of dilated functions." Journal of the European Mathematical Society 017.6 (2015): 1517-1546. <http://eudml.org/doc/277338>.

@article{Aistleitner2015,
abstract = {Upper bounds for GCD sums of the form $\sum _\{k,\{\ell \}=1\}^N\frac\{(\mathrm \{gcd\}(n_k,n_\{\ell \}))^\{2\alpha \}\}\{(n_k n_\{\ell \})^\alpha \}$ are established, where $(n_k)_\{1 \le k \le N\}$ is any sequence of distinct positive integers and $0<\alpha \le 1$; the estimate for $\alpha =1/2$ solves in particular a problem of Dyer and Harman from 1986, and the estimates are optimal except possibly for $\alpha =1/2$. 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 $\mathrm \{Lip\}_\{1/2\}$, 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 $\sum _\{k=1\}^N f(n_k x)$ and the a.e. convergence of $\sum _\{k=1\}^\infty c_k f(n_kx)$ when $f$ is $1$-periodic and of bounded variation or in $\mathrm \{Lip\}_\{1/2\}$.},
author = {Aistleitner, Christoph, Berkes, István, Seip, Kristian},
journal = {Journal of the European Mathematical Society},
keywords = {GCD sums and matrices; Carleson–Hunt inequality; Poisson integral; polydisc; spectral norm; convergence of series of dilated functions; GCD sums and matrices; Carleson-Hunt inequality; Poisson integral; polydisc; spectral norm; convergence of series of dilated functions},
language = {eng},
number = {6},
pages = {1517-1546},
publisher = {European Mathematical Society Publishing House},
title = {GCD sums from Poisson integrals and systems of dilated functions},
url = {http://eudml.org/doc/277338},
volume = {017},
year = {2015},
}

TY - JOUR
AU - Aistleitner, Christoph
AU - Berkes, István
AU - Seip, Kristian
TI - GCD sums from Poisson integrals and systems of dilated functions
JO - Journal of the European Mathematical Society
PY - 2015
PB - European Mathematical Society Publishing House
VL - 017
IS - 6
SP - 1517
EP - 1546
AB - Upper bounds for GCD sums of the form $\sum _{k,{\ell }=1}^N\frac{(\mathrm {gcd}(n_k,n_{\ell }))^{2\alpha }}{(n_k n_{\ell })^\alpha }$ are established, where $(n_k)_{1 \le k \le N}$ is any sequence of distinct positive integers and $0<\alpha \le 1$; the estimate for $\alpha =1/2$ solves in particular a problem of Dyer and Harman from 1986, and the estimates are optimal except possibly for $\alpha =1/2$. 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 $\mathrm {Lip}_{1/2}$, 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 $\sum _{k=1}^N f(n_k x)$ and the a.e. convergence of $\sum _{k=1}^\infty c_k f(n_kx)$ when $f$ is $1$-periodic and of bounded variation or in $\mathrm {Lip}_{1/2}$.
LA - eng
KW - GCD sums and matrices; Carleson–Hunt inequality; Poisson integral; polydisc; spectral norm; convergence of series of dilated functions; GCD sums and matrices; Carleson-Hunt inequality; Poisson integral; polydisc; spectral norm; convergence of series of dilated functions
UR - http://eudml.org/doc/277338
ER -