Upper tails for arithmetic progressions in random subsets

TL;DR

This paper investigates the upper tail probabilities for the count of arithmetic progressions in random subsets, providing optimal exponential bounds and extending results to Schur triples and hypergraph edges.

Contribution

It introduces sharp exponential bounds for upper tails of arithmetic progressions and extends these bounds to Schur triples and hypergraph edge counts.

Findings

Established optimal exponential bounds for upper tails.

Extended bounds to Schur triples.

Generalized results to hypergraph edge counts.

Abstract

We study the upper tail of the number of arithmetic progressions of a given length in a random subset of {1,...,n}, establishing exponential bounds which are best possible up to constant factors in the exponent. The proof also extends to Schur triples, and, more generally, to the number of edges in random induced subhypergraphs of `almost linear' k-uniform hypergraphs.