Small-ball estimates for Hölder centers of the Kloosterman process
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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(sqrt(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 unbiased tensor Bernstein approximation of path functionals. Worst-case statements over rough center classes are distinguished from estimates at a fixed center.
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Research details
- Subjects
- Mathematics · Number Theory and Special Functions
- Keywords
- Kloosterman process, small-ball probabilities, Hölder centers, random Fourier series, Fejér approximation, Wasserstein convergence
- Manuscript date
- 2026-10-19
- Reader reuse
- none
Scope
Uniform-path small-ball orders for admissible Hölder centers, boundary-law thresholds and rough-class distinctions, together with O(sqrt(log log p / log p)) prime-path Wasserstein convergence with both nonzero parameters averaged and uniform shrinking-target counts.
Disclosures
Originality and contribution claims are qualified against inspected prior work and stated source-access/overlap limitations. Classical comparison, approximation, summability and arithmetic inputs are credited. No worldwide priority or optimal-rate claim is made. The critical logarithmic gap is unresolved. AI assistance and the distinction between internal review and external human refereeing are disclosed in the paper. Personal explanations, assessments, internal review records and correspondence are excluded from the public package.
Reproduction
The 36-page manuscript and complete editable source are supplied in a 9-member public package. README.md gives build instructions, dependencies and reproduction scope. Build reproduction is qualified by the compiler, support files and timestamp settings specified in the supplied instructions. The mathematical conclusions have analytic proofs; accompanying finite checks, where present, retain the limited scopes stated in the package.