### 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 introduce a method to compute rigorous component-wise enclosures of discrete convolutions using the fast Fourier transform, the properties of Banach algebras, and interval arithmetic. The purpose of this new approach is to improve the implementation and the applicability of computer-assisted proofs performed in weighed ${\ell}^{1}$ Banach algebras of Fourier/Chebyshev sequences, whose norms are known to be numerically unstable. We introduce some application examples, in particular a rigorous aposteriori...

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,...