"On a conjecture of Leader and Radcliffe related to the Littlewood-Offo" by Tomas Juskevicius
 

Document Type

Lecture

Publication Date

10-29-2014

Abstract

The classical Littlewood-Offord problem is a combinatorial question in geometry that asks for the maximum number subsums of vectors v1 vn Rd of length at least 1 that fall into a xed set A Rd. Erdos proved that the best upper bound in the case d = 1 and A = (xx+2] is n n 2 using Sperners theorem. The problem has a very natural probabilistic formulation. Consider n independent random variables i such that P( i = 1) = 1 2 and let ai 1. Then sup ai x P(a1 1 + +an n (xx+2])=P( 1+ + n ( 11]) In this talk I shall discuss a strong generalization of the latter inequality for arbitrary random variables with only information about their local concentration provided. In particular, a proof of a question posed by Leader and Radcli e will be presented. One of the main ingredients of the proof will be a Sperner type theorem for multisets. Even in the case for sets the proof is new and very elementary.

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