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.