Small-ball estimates for Hölder centers of the Kloosterman process
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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. Classical inputs and inspected prior work are credited.
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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
- No additional permission granted
Scope
Admissible Hölder centers of the Kloosterman limit process and specified bounded symmetric unimodal coefficient laws. Arithmetic convergence averages both prime Kloosterman parameters.
Disclosures
The inspected literature is credited in the manuscript. Classical comparison, approximation, summability and arithmetic inputs are credited. 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
Clean eight-file standard publication package: complete paper PDF, editable LaTeX source including bibliography, three standard-library Python verification scripts, README, reproduction instructions and manifest. Python 3.10 or later. Analytic proofs carry the general results; finite calculations are supporting checks. This is public edition 1; “Revision 3” on the frozen manuscript identifies its local preparation revision, not an earlier public edition.