Tiling the field ${\mathbb{Q}}_p$ of $p$-adic numbers by a function

This study explores the properties of the function which can tile the field ${\mathbb{Q}}_p$ of $p$-adic numbers by translation. It is established that functions capable of tiling ${\mathbb{Q}}_p$ is by translation uniformly locally constancy. As an application, in the field ${\mathbb{Q}}_p$, we addressed the question posed by H. Leptin and D. Müller, providing the necessary and sufficient conditions for a discrete set to correspond to a uniform partition of unity. The study also connects these tiling properties to the Fuglede conjecture, which states that a measurable set is a tile if and only if it is spectral. The paper concludes by characterizing the structure of tiles in ${\mathbb{Q}}_p\times \mathbb{Z}\mathrm{/}2\mathbb{Z}$, proving that they are spectral sets.