We are going to prove that if \(E \subset {\Bbb Z}_p^2\), \(p\) prime, then \(L^2(E)\) has an orthogonal basis of characters if and only if \(E\) tiles \({\Bbb Z}_p^2\) by translation. The (fairly elementary) proof has an analytic, combinatorial and number theoretic components. This is joint work with Azita Mayeli and Jonathan Pakianathan.