Small-ball estimates for Hölder centers of the Kloosterman process

Overview

Abstract

We prove uniform-norm small-ball bounds of logarithmic order ε⁻² about every admissible 1/2-Hölder center of the Kloosterman limit process, including boundary coefficient and endpoint values. Graded Fejér approximation removes two logarithmic losses from an inspected earlier lower estimate. We determine worst-case power orders over admissible Hölder classes, give an explicit O(log log p/√log p) uniform-path Wasserstein convergence rate for prime Kloosterman paths with both parameters averaged, and derive uniform shrinking-target counts. Extensions to bounded symmetric unimodal coefficient laws include sharp noncritical boundary power thresholds and a fixed-center obstruction at equality. The critical logarithmic gap remains unresolved. Classical inputs and inspected prior work are credited; no priority or optimal-rate claim is made.

Keywords Kloosterman process · small-ball probabilities · Hölder centers · random Fourier series · Fejér approximation · Wasserstein convergence