# GCD sums from Poisson integrals and systems of dilated functions

Christoph Aistleitner; István Berkes; Kristian Seip

Journal of the European Mathematical Society (2015)

- Volume: 017, Issue: 6, page 1517-1546
- ISSN: 1435-9855

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topAistleitner, 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 -

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