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
- 1 · 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. 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
Scope of the work
Admissible Hölder centers of the Kloosterman limit process and specified bounded symmetric unimodal coefficient laws. Arithmetic convergence averages both prime Kloosterman parameters. Critical logarithmic upper/lower exponent gap remains open; no Riemann-hypothesis, universal optimal-rate or practical-efficiency claim.
Reproduction notes
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.
The manuscript
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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.
A stable scholarly record
Cite this edition
Scott Schubr. (2026). Small-ball estimates for Hölder centers of the Kloosterman process (Version 1). Luminary Research Archive. https://research.luminaryoddities.com/small-ball-estimates-holder-centers-kloosterman-process/v/1
BibTeX
@techreport{luminary_small_ball_estimates_holder_centers_kloosterman_process_v1,
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 1},
url = {https://research.luminaryoddities.com/small-ball-estimates-holder-centers-kloosterman-process/v/1}
}Loading discussion…