This research demonstrates optimal lower bounds in anti-concentration probabilities for Rademacher sums, highlighting independent variables and uniform signs.
We consider lower bounds on anti‐concentration probabilities of the form , where is a Rademacher sum; are positive constants normalised so , and are independent and uniform signs . We determine the value of for all values , giving a partial answer to a question by Keller and Klein. In particular, for , we prove the optimal lower bound , which improves on a sequence of results by Burkholder, Oleszkiewicz, Hitczenko and Kwapień, and Dvořák and Klein, and confirms a 1994 conjecture of Hitczenko and Kwapień. We also discuss some higher‐dimensional variants of the problem.
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Hollom et al. (2025) studied this question.
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