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Combinatorial anti-concentration inequalities, with applications

Part of: Combinatorial probability Graph theory Extremal combinatorics

Published online by Cambridge University Press:  30 June 2021

JACOB FOX ,
MATTHEW KWAN  and
LISA SAUERMANN
JACOB FOX
Affiliation:
Department of Mathematics, Stanford University, 450 Jane Stanford Way, Building 380, Stanford, CA 94305-2125, U.S.A. e-mails: jacobfox@stanford.edu, mattkwan@stanford.edu, Isauerma@stanford.edu
MATTHEW KWAN
Affiliation:
Department of Mathematics, Stanford University, 450 Jane Stanford Way, Building 380, Stanford, CA 94305-2125, U.S.A. e-mails: jacobfox@stanford.edu, mattkwan@stanford.edu, Isauerma@stanford.edu
LISA SAUERMANN
Affiliation:
Department of Mathematics, Stanford University, 450 Jane Stanford Way, Building 380, Stanford, CA 94305-2125, U.S.A. e-mails: jacobfox@stanford.edu, mattkwan@stanford.edu, Isauerma@stanford.edu
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Abstract

We prove several different anti-concentration inequalities for functions of independent Bernoulli-distributed random variables. First, motivated by a conjecture of Alon, Hefetz, Krivelevich and Tyomkyn, we prove some “Poisson-type” anti-concentration theorems that give bounds of the form 1/e + o(1) for the point probabilities of certain polynomials. Second, we prove an anti-concentration inequality for polynomials with nonnegative coefficients which extends the classical Erdős–Littlewood–Offord theorem and improves a theorem of Meka, Nguyen and Vu for polynomials of this type. As an application, we prove some new anti-concentration bounds for subgraph counts in random graphs.

MSC classification

Primary: 05D40: Probabilistic methods 60C05: Combinatorial probability 05C80: Random graphs
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 171 , Issue 2 , September 2021 , pp. 227 - 248
Copyright
© The Author(s), 2021. Published by Cambridge University Press on behalf of Cambridge Philosophical Society

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Footnotes

†

Research supported by a Packard Fellowship and by NSF Award DMS-1855635.

‡

Research supported in part by SNSF project 178493.

References

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