For a locally compact abelian group $G$, Fuglede's
conjecture states that a Borel set is spectral if and only if it tiles
the group $G$ by translation. In the case $G=\mathbb{R}^n$, it have
been studied for long time since Fuglede formulated this conjecture in
1974. It is proved to be false for $n\ge 3$ but it is still open for
$n=1,2$. With A.Fan, S.Fan and L.Liao, we consider the case
$G=\mathbb{Q}_p$ the field of p-adic numbers and give an affirmative
answer to the conjecture in this case. Moreover, we prove that the
spectral sets in $\mathbb{Q}_p$ are compact open up to a null set.
With A.Fan and S.Fan, we give a geometric criterion of spectral
compact open sets, which is called $p$-homogeneity. In this talk, I
will first give an overview of Fuglede's conjecture and then talk
about $p$-homogeneity.