Functions tiling simultaneously with two arithmetic progressions

TL;DR

This paper investigates the conditions under which measurable functions can tile the real line simultaneously with two different arithmetic progressions, revealing how the arithmetic relationships between the progressions influence tiling levels and support size.

Contribution

It provides sharp results characterizing possible tiling levels and minimal support measure based on the arithmetic properties of the progressions and tiling levels.

Findings

Tiling levels depend on rational independence of the progressions.

Minimal support measure varies with arithmetic relations between parameters.

Results are sharp and depend on the rationality of lpha, eta.

Abstract

We consider measurable functions $f$ on $\mathbb{R}$ that tile simultaneously by two arithmetic progressions $\alpha \mathbb{Z}$ and $\beta \mathbb{Z}$ at respective tiling levels $p$ and $q$. We are interested in two main questions: what are the possible values of the tiling levels $p,q$, and what is the least possible measure of the support of $f$? We obtain sharp results which show that the answers depend on arithmetic properties of $\alpha, \beta$ and $p,q$, and in particular, on whether the numbers $\alpha, \beta$ are rationally independent or not.