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

In this note we provide a simple explicit example of a discrete set $\mathrm{\Lambda }\subset \mathbb{R}$ of unbounded density such that the Fourier transform of its counting measure ${\widehat{\delta }}_{\mathrm{\Lambda }}$ is a tempered distribution of the form $c{\delta }_0^{\mathrm{\prime }\mathrm{\prime }}+\mu$ 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.