Multivariate Davenport series

TL;DR

This paper investigates the regularity, multifractal properties, and open problems of multivariate Davenport series, which are extensions of classical Davenport series involving multidimensional Fourier-like sums.

Contribution

It introduces the multivariate Davenport series, analyzes their Sobolev and pointwise regularity, and explores their multifractal characteristics, providing new insights into their mathematical structure.

Findings

Determined the Sobolev regularity of multivariate Davenport series

Analyzed their pointwise regularity and multifractal spectrum

Outlined open problems for future research on these series

Abstract

We consider series of the form $\sum a_n \{n\cdot x\}$, where $n\in\Z^{d}$ and $\{x\}$ is the sawtooth function. They are the natural multivariate extension of Davenport series. Their global (Sobolev) and pointwise regularity are studied and their multifractal properties are derived. Finally, we list some open problems which concern the study of these series.