Small-ball estimates for Hölder centers of the Kloosterman process

Abstract

We prove uniform-norm small-ball estimates with negative logarithm of order ε⁻² around every admissible 1/2-Hölder center of the Kloosterman limit process, including boundary coefficients and endpoints. Graded Fejér approximation removes logarithmic losses from earlier lower estimates. We also prove an O(log log p/√(log p)) Wasserstein convergence rate for the entire prime Kloosterman path with both nonzero parameters averaged, in the uniform path metric, and deduce uniform shrinking-target counts. For fixed admissible γ-Hölder classes, the worst-case small-ball power is max(2,1/γ). Extensions to bounded symmetric unimodal coefficient laws identify how center roughness and boundary decay of the density jointly determine approximation costs. For a flat-edge family we obtain sharp noncritical power thresholds and fixed-center obstructions at the threshold. At the critical law we establish upper and lower logarithmic bounds. The sharp small-ball exponent is obtained by paying the random tail and the coefficient-boundary margins separately; the arithmetic argument keeps the representation-degree dependence in classical trace estimates and uses growing moments and weighted polynomial approximation. Worst-case statements over rough center classes are distinguished from estimates at a fixed center.