Boolean Small-Ball Inequalities for Discrepancy Theory
Abstract
We prove new small-ball inequalities for boolean matrix-series. The leading example is 𝔼s[det(I−S2)β 1{|S|<1}]≥ e−O(βτ), which holds for boolean matrix-series S=∑i siAi formed using symmetric matrices A1,…,An and uniformly random signs s∈{±1}n. Specifically, this inequality holds for all β≥1 with τ=∑iTr Ai2, as soon as the maximum of (Tr Ai2)i=1n and a certain variance term are bounded above by universal constants. The proof combines the Gaussian reciprocal estimate of (Akbas and Sra 2026), the directional-variation signing theorem of (Guo, Fang, and Lu 2026), and a replica argument that turns existence into a Gibbs law on good signings. Most notably, boolean small-ball delivers a new, interlacing-free proof of Kadison-Singer (most general case); it also recovers Matrix Spencer and Komlós as quick corollaries, while yielding more than six almost immediate proofs of an assortment of discrepancy theoretic problems.
Related
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BibTeX
@misc{akbas2026boolean,
title = {{Boolean Small-Ball Inequalities for Discrepancy Theory}},
author = {Emrullah Akbas and Suvrit Sra},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.20785},
eprint = {2609.20785},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.20785},
}
RIS
TY - UNPB
TI - Boolean Small-Ball Inequalities for Discrepancy Theory
AU - Emrullah Akbas
AU - Suvrit Sra
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.20785
UR - https://arxiv.org/abs/2609.20785
AB - We prove new small-ball inequalities for boolean matrix-series. The leading example is 𝔼_s[det(I−S^2)^β 1_{|S|<1}]≥ e^−O(βτ), which holds for boolean matrix-series S=∑_i s_iA_i formed using symmetric matrices A_1,…,A_n and uniformly random signs s∈{±1}^n. Specifically, this inequality holds for all β≥1 with τ=∑_iTr A_i^2, as soon as the maximum of (Tr A_i^2)_i=1^n and a certain variance term are bounded above by universal constants. The proof combines the Gaussian reciprocal estimate of (Akbas and Sra 2026), the directional-variation signing theorem of (Guo, Fang, and Lu 2026), and a replica argument that turns existence into a Gibbs law on good signings. Most notably, boolean small-ball delivers a new, interlacing-free proof of Kadison-Singer (most general case); it also recovers Matrix Spencer and Komlós as quick corollaries, while yielding more than six almost immediate proofs of an assortment of discrepancy theoretic problems.
ER -