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arXiv · 2211.16058

Functions tiling simultaneously with two arithmetic progressions

Mark Mordechai Etkind·Nir Lev
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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.

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Mark Mordechai Etkind, Nir Lev. 2022-11-29. Functions tiling simultaneously with two arithmetic progressions. https://doi.org/10.1112/plms.12570

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