Upper tails for arithmetic progressions in a random set

TL;DR

This paper determines the asymptotic behavior of the probability that the number of k-term arithmetic progressions in a random subset exceeds its expectation by a certain factor, using advanced large deviation principles.

Contribution

It provides the first precise asymptotic estimates for large deviations of arithmetic progressions in sparse random sets, answering a key open question.

Findings

Established asymptotics of large deviation probabilities for arithmetic progressions

Extended the nonlinear large deviation principle to this combinatorial setting

Complemented previous bounds by Warnke with exact asymptotics

Abstract

Let $X_k$ denote the number of $k$-term arithmetic progressions in a random subset of $\mathbb{Z}/N\mathbb{Z}$ or $\{1, \dots, N\}$ where every element is included independently with probability $p$. We determine the asymptotics of $\log \mathbb{P}(X_k \ge (1+\delta) \mathbb{E} X_k)$ (also known as the large deviation rate) where $p \to 0$ with $p \ge N^{-c_k}$ for some constant $c_k > 0$, which answers a question of Chatterjee and Dembo. The proofs rely on the recent nonlinear large deviation principle of Eldan, which improved on earlier results of Chatterjee and Dembo. Our results complement those of Warnke, who used completely different methods to estimate, for the full range of $p$, the large deviation rate up to a constant factor.