### A Cantor-Lebesgue theorem for double trigonometric series

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The aim of this paper is to obtain sharp estimates from below of the measure of the set of divergence of the m-fold Fourier series with respect to uniformly bounded orthonormal systems for the so-called G-convergence and λ-restricted convergence. We continue the study begun in a previous work.

We establish a connection between the L² norm of sums of dilated functions whose jth Fourier coefficients are $\left({j}^{-\alpha}\right)$ 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.

Denote by ${f}_{ss}(x,y)$ the sum of a double sine series with nonnegative coefficients. We present necessary and sufficient coefficient conditions in order that ${f}_{ss}$ 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.

We derive new upper bounds for the densities of measurable sets in ${\mathbb{R}}^{n}$ 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 $2,\cdots ,24$. This gives new lower bounds for the measurable chromatic number in dimensions $3,\cdots ,24$. We apply it to get a short proof of a variant of a recent result of Bukh which in turn generalizes theorems of Furstenberg,...

Upper bounds for GCD sums of the form ${\sum}_{k,\ell =1}^{N}\frac{{\left(\mathrm{gcd}({n}_{k},{n}_{\ell})\right)}^{2\alpha}}{{\left({n}_{k}{n}_{\ell}\right)}^{\alpha}}$ are established, where ${\left({n}_{k}\right)}_{1\le k\le N}$ 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...