PreprintCC BY 4.0arXiv
A Counterexample to Belinsky's Conjecture on Cesàro Means at Lebesgue Points
Abstract
In 1997, Belinsky conjectured that, for convex subsequences, the logarithmic growth condition of Carleson, Trigub, and Zagorodni\uı is necessary and sufficient for the arithmetic means of subsequential Fourier partial sums to converge at every Lebesgue point of every integrable function. We disprove the sufficiency part of this conjecture. More precisely, we construct a strictly convex increasing sequence (am) satisfying am≤ 7m8 and a function f∈ L1(𝕋) for which $0$ is a Lebesgue point, f(0)=0, and the means m−1∑k=1m Sakf(0) are unbounded.
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BibTeX
@misc{goginava2026counterexample,
title = {{A Counterexample to Belinsky's Conjecture on Cesàro Means at Lebesgue Points}},
author = {Ushangi Goginava},
howpublished = {arXiv},
year = {2026},
month = aug,
doi = {10.48550/arxiv.2608.30575},
eprint = {2608.30575},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2608.30575},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - A Counterexample to Belinsky's Conjecture on Cesàro Means at Lebesgue Points
AU - Ushangi Goginava
PY - 2026
DA - 2026/08/31
PB - arXiv
DO - 10.48550/arxiv.2608.30575
UR - https://arxiv.org/abs/2608.30575
AB - In 1997, Belinsky conjectured that, for convex subsequences, the logarithmic growth condition of Carleson, Trigub, and Zagorodni\uı is necessary and sufficient for the arithmetic means of subsequential Fourier partial sums to converge at every Lebesgue point of every integrable function. We disprove the sufficiency part of this conjecture. More precisely, we construct a strictly convex increasing sequence (a_m) satisfying a_m≤ 7m^8 and a function f∈ L^1(𝕋) for which $0$ is a Lebesgue point, f(0)=0, and the means m^−1∑_k=1^m S_a_kf(0) are unbounded.
ER -