{"ID":23612725,"CreatedAt":"2026-09-18T06:30:02.82110233Z","UpdatedAt":"2026-09-18T06:30:02.82110233Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2609.20785","arxiv_id":"2609.20785","title":"Boolean Small-Ball Inequalities for Discrepancy Theory","abstract":"We prove new small-ball inequalities for boolean matrix-series. The leading example is $\\mathbb E_s[{\\text{det}(I-S^2)^β\\,\\mathbf 1_{\\{\\|S\\|\u003c1\\}}}]\\ge e^{-O(βτ)}$, which holds for boolean matrix-series $S=\\sum_i s_iA_i$ formed using symmetric matrices $A_1,\\dots,A_n$ and uniformly random signs $s\\in\\{\\pm1\\}^n$. Specifically, this inequality holds for all $β\\ge1$ with $τ=\\sum_i\\text{Tr} A_i^2$, as soon as the maximum of $(\\text{Tr} A_i^2)_{i=1}^n$ 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.","short_abstract":"We prove new small-ball inequalities for boolean matrix-series. The leading example is $\\mathbb E_s[{\\text{det}(I-S^2)^β\\,\\mathbf 1_{\\{\\|S\\|\u003c1\\}}}]\\ge e^{-O(βτ)}$, which holds for boolean matrix-series $S=\\sum_i s_iA_i$ formed using symmetric matrices $A_1,\\dots,A_n$ and uniformly random signs $s\\in\\{\\pm1\\}^n$. Specifi...","url_abs":"https://arxiv.org/abs/2609.20785","url_pdf":"https://arxiv.org/pdf/2609.20785v1","authors":"[\"Emrullah Akbas\",\"Suvrit Sra\"]","published":"2026-09-17T17:52:14Z","proceeding":"math.PR","tasks":"[\"math.PR\",\"math.CO\",\"math.FA\"]","methods":"[]","has_code":false}