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Stability of Szarek's inequality with best constant
Abstract
Let ε1,…,εn be independent Rademacher random variables. We settle the best constant c such that 𝔼|∑j=1n ajεj| ≥ 1/√2 +c|a−(e1+e2)/√2| holds for every n≥ 2 and every unit vector a∈ℝn with a1≥⋯≥ an≥ 0. We prove that the optimal constant is c=(2−√2)3/2/8, with equality attained at a=1/2(1,1,1,1,0,…,0) for n≥ 4. This completes a line of research by De-Diakonikolas-Servedio, Eskenazis-Nayar-Tkocz and Fang-Wang.
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BibTeX
@misc{xie2026stability,
title = {{Stability of Szarek's inequality with best constant}},
author = {Xinyuan Xie},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.07526},
eprint = {2610.07526},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.07526},
}
RIS
TY - UNPB
TI - Stability of Szarek's inequality with best constant
AU - Xinyuan Xie
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.07526
UR - https://arxiv.org/abs/2610.07526
AB - Let ε_1,…,ε_n be independent Rademacher random variables. We settle the best constant c such that 𝔼|∑_j=1^n a_jε_j| ≥ 1/√2 +c|a−(e_1+e_2)/√2| holds for every n≥ 2 and every unit vector a∈ℝ^n with a_1≥⋯≥ a_n≥ 0. We prove that the optimal constant is c=(2−√2)^3/2/8, with equality attained at a=1/2(1,1,1,1,0,…,0) for n≥ 4. This completes a line of research by De-Diakonikolas-Servedio, Eskenazis-Nayar-Tkocz and Fang-Wang.
ER -