We proved that Fuglede's conjecture concerning spectral sets and tilings holds in the field of $p$-adic numbers, i.e. a Borel set of positive and finite Haar measure is a spectral set if and only if it tiles the space by translation, although the conjecture remains open in the field of real numbers. Our study is based on the investigation of a convolution equation of the form $f^* \mu =1$, where $\mu$ is a measure supported by a discrete set and $f$ is a non-negative integrable function. I. J. Schoenberg's result concerning the $p^n$-th roots of unity plays a crucial role. It is a joint work with Ai-Hua Fan, Lingmin Liao and Ruxi Shi
