Space-filling curves and small-ball probabilities for the Kloosterman process
Abstract
We construct explicitly a space-filling curve in the support of the limiting Kloosterman process, using the planar partial-sum curve of the Rudin--Shapiro sequence. The qualitative existence statement was previously attributed to J. Bober by Kowalski. The resulting curve is a closed, 1/2-Hölder continuous loop whose image contains a specified triangle of area 1/128. Its Fourier coefficients are purely imaginary and satisfy |f̂(h)|≤(2π|h|)−1 for h≠ 0. We also prove a uniform lower bound for small balls centred at closed 1/2-Hölder curves with a fixed coefficient bound, of the form exp{−Cε−2log2(2/ε)}. Known functional convergence then gives a quantitative lower bound for the limiting proportion of two-parameter Kloosterman paths approximating the constructed loop.
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BibTeX
@misc{kyselov2026spacefilling,
title = {{Space-filling curves and small-ball probabilities for the Kloosterman process}},
author = {Danyil Kyselov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22645},
eprint = {2610.22645},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22645},
}
RIS
TY - UNPB
TI - Space-filling curves and small-ball probabilities for the Kloosterman process
AU - Danyil Kyselov
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.22645
UR - https://arxiv.org/abs/2610.22645
AB - We construct explicitly a space-filling curve in the support of the limiting Kloosterman process, using the planar partial-sum curve of the Rudin--Shapiro sequence. The qualitative existence statement was previously attributed to J. Bober by Kowalski. The resulting curve is a closed, 1/2-Hölder continuous loop whose image contains a specified triangle of area 1/128. Its Fourier coefficients are purely imaginary and satisfy |f̂(h)|≤(2π|h|)^−1 for h≠ 0. We also prove a uniform lower bound for small balls centred at closed 1/2-Hölder curves with a fixed coefficient bound, of the form exp{−Cε^−2log^2(2/ε)}. Known functional convergence then gives a quantitative lower bound for the limiting proportion of two-parameter Kloosterman paths approximating the constructed loop.
ER -