A new example of crystalline-type measure with unit masses and an application to tiling of the real line by translates of a function

Abstract

In this note we provide a simple explicit example of a discrete set $Λ\subset\mathbb{R}$ of unbounded density such that the Fourier transform of its counting measure $\widehatδ_Λ$ is a tempered distribution of the form $cδ_0'' + μ$ where $μ$ is a discrete measure supported by a union of three arithmetic progressions. This example is then used to construct a translational tiling of unbounded density of the real line by a function with compact support.

Disclosure

“CRYSTALLINE-TYPE MEASURE AND AN APPLICATION TO TRANSLATIONAL TILING 5 The TikZ code for Figure 2.1 below was generated by ChatGPT. Apart from that, the text of the paper was written manually by the author without using LLMs (there- fore at least all possible typos are the author’s contribution). The author takes full responsibility for the mathematical content of the”

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2026-08-23T05:07:18.392708Z

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