Distribution of Kloosterman paths to high prime power moduli

Abstract

We consider the distribution of polygonal paths joining the partial sums of normalized Kloosterman sums modulo an increasingly high power p n p^n of a fixed odd prime p p , a pure depth-aspect analogue of theorems of Kowalski–Sawin and Ricotta–Royer–Shparlinski. We find that this collection of Kloosterman paths naturally splits into finitely many disjoint ensembles, each of which converges in law as n → ∞ n\to \infty to a distinct complex valued random continuous function. We further find that the random series resulting from gluing together these limits for every p p converges in law as p → ∞ p\to \infty , and that paths joining partial Kloosterman sums acquire a different and universal limiting shape after a modest rearrangement of terms. As the key arithmetic input we prove, using the p p -adic method of stationary phase including highly singular cases, that complete sums of products of arbitrarily many Kloosterman sums to high prime power moduli exhibit either power savings or power alignment in shifts of arguments.