Critical convergence and Hausdorff measures for generalized Flint Hills series
Abstract
We study the generalized Flint Hills series ℱs,t(x)=∑n≥1 n−s|sin(πnx)|−t for s>0 and t>1. An explicit comparison with a series over continued-fraction denominators yields the Hausdorff dimension min{1,2t/(s+t)} of its divergence set. At each critical exponent τ=1+s/t>2 we construct numbers of irrationality exponent τ realizing both convergence and divergence; the convergent examples establish Meiburg's conjecture in the range t>1. Both parts of the critical fibre have Hausdorff dimension 2/τ. For s>t and hκ(r)=r2/τ(log(1/r))κ we prove that the divergence set has zero hκ-measure for κ<−1 and infinite measure for κ≥−1. The divergent part of the critical fibre satisfies the same law, whereas its convergent part has infinite measure for every κ. The key estimate selects a rapidly growing subsequence of convergent denominators and gives a double-logarithmic bound on the approximation error. The convergence of the classical Flint Hills series ∑n≥1(n3sin2 n)−1 remains undecided.
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BibTeX
@misc{dan2026critical,
title = {{Critical convergence and Hausdorff measures for generalized Flint Hills series}},
author = {Yuya Dan},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.12338},
eprint = {2609.12338},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.12338},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Critical convergence and Hausdorff measures for generalized Flint Hills series
AU - Yuya Dan
PY - 2026
DA - 2026/09/11
PB - arXiv
DO - 10.48550/arxiv.2609.12338
UR - https://arxiv.org/abs/2609.12338
AB - We study the generalized Flint Hills series ℱ_s,t(x)=∑_n≥1 n^−s|sin(πnx)|^−t for s>0 and t>1. An explicit comparison with a series over continued-fraction denominators yields the Hausdorff dimension min{1,2t/(s+t)} of its divergence set. At each critical exponent τ=1+s/t>2 we construct numbers of irrationality exponent τ realizing both convergence and divergence; the convergent examples establish Meiburg's conjecture in the range t>1. Both parts of the critical fibre have Hausdorff dimension 2/τ. For s>t and h_κ(r)=r^2/τ(log(1/r))^κ we prove that the divergence set has zero h_κ-measure for κ<−1 and infinite measure for κ≥−1. The divergent part of the critical fibre satisfies the same law, whereas its convergent part has infinite measure for every κ. The key estimate selects a rapidly growing subsequence of convergent denominators and gives a double-logarithmic bound on the approximation error. The convergence of the classical Flint Hills series ∑_n≥1(n^3sin^2 n)^−1 remains undecided.
ER -