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arXiv · 2009.09410

Tiling by translates of a function: results and open problems

Mihail N. Kolountzakis·Nir Lev
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Abstract

We say that a function $f \in L^1(\mathbb{R})$ tiles at level $w$ by a discrete translation set $\Lambda \subset \mathbb{R}$, if we have $\sum_{\lambda \in \Lambda} f(x-\lambda)=w$ a.e. In this paper we survey the main results, and prove several new ones, on the structure of tilings of $\mathbb{R}$ by translates of a function. The phenomena discussed include tilings of bounded and of unbounded density, uniform distribution of the translates, periodic and non-periodic tilings, and tilings at level zero. Fourier analysis plays an important role in the proofs. Some open problems are also given.

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BibTeXRIS

Mihail N. Kolountzakis, Nir Lev. 2020-09-20. Tiling by translates of a function: results and open problems. https://doi.org/10.19086/da.28122

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