Research Article · Preprint Mathematics
Small-ball estimates for Hölder centers of the Kloosterman process
- Primary subject
- Mathematics
- Subject subfields
- Number Theory and Special Functions
- Paper type
- Research Article
- Manuscript stage
- Preprint
- Manuscript
- Archived
- Edition
- 2 · Permanent link
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. A graded Fejér approximation removes two logarithmic losses from an inspected earlier lower estimate. For every 0<γ≤1, the worst probability over a fixed admissible γ-Hölder class has negative logarithm of order ε^(-max(2,1/γ)). We also prove an explicit O(log log p/√(log p)) Wasserstein convergence rate, in the uniform path metric, for prime Kloosterman paths with both parameters averaged; this yields uniform shrinking-target counts. The argument quantifies classical shifted-correlation and approximation tools. For other bounded symmetric unimodal coefficient laws, a fractional Fejér method gives a sufficient boundary-integrability condition. For densities proportional to exp[-A(2-|x|)^(-a)], the uniform lower order ε^(-1/γ), 0<γ≤1/2, holds exactly when a<1/γ. A fixed admissible multiscale boundary center proves failure at equality. At γ=1/2,a=2, we give a uniform ε^(-2)log^3(e/ε) lower-probability exponent, complementing a fixed-center ε^(-2)log^2(1/ε) upper-probability exponent. For 0<γ<1, the noncritical worst-case power order is ε^(-max(2,1/γ,a)); for γ>1/2 this holds for every a>0, including a=2. All comparison, summability and arithmetic machinery is credited to its classical sources.
Keywords Kloosterman process · small-ball probabilities · Hölder centers · random Fourier series · Fejér approximation · Wasserstein convergence
Scope of the work
Uniform Kloosterman-process small-ball bounds for admissible Hölder centers, exact worst-case power orders and boundary-law thresholds, with explicit prime-path Wasserstein convergence and shrinking-target counts under the stated averaging.
Reproduction notes
The paper has 35 pages. The editable-source archive contains 8 files. Compile Manuscript.tex. The standard-library package, finite-identity and Revision 3 scripts check file identities and the separately described exact/floating finite controls in a fresh output directory.
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Author 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.
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Cite this edition
Scott Schubr. (2026). Small-ball estimates for Hölder centers of the Kloosterman process (Version 2). Luminary Research Archive. https://research.luminaryoddities.com/small-ball-estimates-holder-centers-kloosterman-process/v/2
BibTeX
@techreport{luminary_small_ball_estimates_holder_centers_kloosterman_process_v2,
title = {Small-ball estimates for Hölder centers of the Kloosterman process},
author = {Scott Schubr},
year = {2026},
institution = {Luminary Research Archive},
type = {Research Article},
note = {Version 2},
url = {https://research.luminaryoddities.com/small-ball-estimates-holder-centers-kloosterman-process/v/2}
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