On the missing log in upper tail estimates

TL;DR

This paper introduces a combinatorial sparsification technique based on the BK-inequality to recover the missing logarithmic factor in upper tail estimates for certain random variables, with applications in additive combinatorics.

Contribution

It presents a novel method that refines upper tail bounds by closing the logarithmic gap in existing inductive approaches.

Findings

Achieves sharp upper tail estimates for arithmetic progressions and Schur triples.

Provides a new combinatorial approach applicable to various additive structures.

Improves the precision of probabilistic bounds in additive combinatorics.

Abstract

In the late 1990s, Kim and Vu pioneered an inductive method for showing concentration of certain random variables X. Shortly afterwards, Janson and Ruci{\'n}ski developed an alternative inductive approach, which often gives comparable results for the upper tail Pr(X \ge (1+\eps) E[X]). In some cases, both methods yield upper tail estimates which are best possible up to a logarithmic factor in the exponent, but closing this narrow gap has remained a technical challenge. In this paper we present a BK-inequality based combinatorial sparsification idea that can recover this missing logarithmic term in the upper tail. As an illustration, we consider random subsets of the integers {1,...,n}, and prove sharp upper tail estimates for various objects of interest in additive combinatorics. Examples include the number of arithmetic progressions, Schur triples, additive quadruples, and (r,s)-sums.