Functions tiling simultaneously with two arithmetic progressions

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.

To appear in the Proceedings of the London Mathematical Society

References in corpus (4)

• Tiling by translates of a function: results and open problems
• An example concerning Fourier analytic criteria for translational tiling
• Doubly stochastic arrays with small support
• Support of extremal doubly stochastic arrays