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 $\Lambda\subset\mathbb{R}$ of unbounded density such that the Fourier transform of its counting measure $\widehat{\delta}_\Lambda$ is a tempered distribution of the form $c\delta_0'' + \mu$ where $\mu$ 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.