Abstract: A Borel subset in R^d of positive and finite Lebesgue measure is called a spectral set if the spece of square integrable functions on it admits an orthogonal basis consisting of exponential functions. Fuglede conjecture (1974) states that a Borel set is a spectral set if and only if it tiles the whole space R^d by translation. Though the conjecture is false for higher dimensions, it is still open for R^1 and R^2. We prove the Fuglede conjecture in the one dimensional p-adic space, i.e., a Borel set of positive and finite Haar measure in the field Q_p of p-adic numbers is a spectral set if and only if it tiles Q_p by translation. This is a joint work with Ai-Hua Fan, Shilei Fan and Ruxi Shi.
