Tiling by translates of a function: results and open problems

Mihail N. Kolountzakis; Nir Lev

doi:10.19086/da.28122

Tiling by translates of a function: results and open problems

Mihail N. Kolountzakis, Nir Lev

Department of Mathematics

Bar-Ilan University

Research output: Contribution to journal › Article › peer-review

Abstract

We say that a function (Formula Presented) tiles at level w by a discrete translation set ⋀⊂R, if we have (Formula Presented) In this paper we survey the main results, and prove several new ones, on the structure of tilings of 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.

Original language	English
Article number	12
Journal	Discrete Analysis
Volume	2021
DOIs	https://doi.org/10.19086/da.28122
State	Published - 2021

Keywords

Quasicrystals
Spectral gap
Tiling
Translates
Uncertainty principle

All Science Journal Classification (ASJC) codes

Algebra and Number Theory
Geometry and Topology
Discrete Mathematics and Combinatorics

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10.19086/da.28122

Cite this

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AU - Kolountzakis, Mihail N.

AU - Lev, Nir

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AB - We say that a function (Formula Presented) tiles at level w by a discrete translation set ⋀⊂R, if we have (Formula Presented) In this paper we survey the main results, and prove several new ones, on the structure of tilings of 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.

KW - Quasicrystals

KW - Spectral gap

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KW - Translates

KW - Uncertainty principle

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M3 - مقالة

SN - 2397-3129

VL - 2021

JO - Discrete Analysis

JF - Discrete Analysis

M1 - 12

ER -

Tiling by translates of a function: results and open problems

Abstract

Keywords

All Science Journal Classification (ASJC) codes

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