Pair Correlation of Squares in p-Adic Fields

Let p be an odd prime number, K a p-adic field of degree r over mathbf{Q}p, O the ring of integers in K, B = {beta1,..., betar} an integral basis of K over mathbf{Q}p, u a unit in O and consider sets of the form mathcal{N}={n1 beta1 + ... + nr betar : 1 <= nj <= Nj, 1 <= j <= r}. We show under certain growth conditions that the pair correlation of {uz2 : z in mathcal{N}} becomes Poissonian.