Pair Correlation of Squares in $p$-Adic Fields

[摘要] 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 $K$, $B = {eta_1,dots,eta_r}$ an integral basis of $K$ over $mathbf{Q}_p$, $u$ a unit in $O$and consider sets of the form $mathcal{N}={n_1eta_1+cdots+n_reta_r:1leq n_jleq N_j, 1leq jleq r}$. We show under certain growthconditions that the pair correlation of ${uz^2:zinmathcal{N}}$ becomes Poissonian.

[学科分类] 数学（综合）