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Convergence of series of dilated functions and spectral norms of GCD matrices

Christoph Aistleitner, István Berkes, Kristian Seip, Michel Weber (2015)

Acta Arithmetica

We establish a connection between the L² norm of sums of dilated functions whose jth Fourier coefficients are ( j - α ) for some α ∈ (1/2,1), and the spectral norms of certain greatest common divisor (GCD) matrices. Utilizing recent bounds for these spectral norms, we obtain sharp conditions for the convergence in L² and for the almost everywhere convergence of series of dilated functions.

Double sine series with nonnegative coefficients and Lipschitz classes

Vanda Fülöp (2006)

Colloquium Mathematicae

Denote by f s s ( x , y ) the sum of a double sine series with nonnegative coefficients. We present necessary and sufficient coefficient conditions in order that f s s belongs to the two-dimensional multiplicative Lipschitz class Lip(α,β) for some 0 < α ≤ 1 and 0 < β ≤ 1. Our theorems are extensions of the corresponding theorems by Boas for single sine series.

Fourier analysis, linear programming, and densities of distance avoiding sets in n

Fernando Mário de Oliveira Filho, Frank Vallentin (2010)

Journal of the European Mathematical Society

We derive new upper bounds for the densities of measurable sets in n which avoid a finite set of prescribed distances. The new bounds come from the solution of a linear programming problem. We apply this method to obtain new upper bounds for measurable sets which avoid the unit distance in dimensions 2 , , 24 . This gives new lower bounds for the measurable chromatic number in dimensions 3 , , 24 . We apply it to get a short proof of a variant of a recent result of Bukh which in turn generalizes theorems of Furstenberg,...

GCD sums from Poisson integrals and systems of dilated functions

Christoph Aistleitner, István Berkes, Kristian Seip (2015)

Journal of the European Mathematical Society

Upper bounds for GCD sums of the form k , = 1 N ( gcd ( n k , n ) ) 2 α ( n k n ) α are established, where ( n k ) 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...

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