Upper tails for arithmetic progressions in a random set

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.

References in corpus (2)

• Upper tails and independence polynomials in random graphs
• Upper tails for arithmetic progressions in random subsets

Cited by in corpus (6)

• Gaussian width bounds with applications to arithmetic progressions in random settings
• Local limit theorems for subgraph counts
• Maximising the number of solutions to a linear equation in a set of integers
• A large deviation principle for block models
• The maximal number of $3$-term arithmetic progressions in finite sets in different geometries
• Preferential Attachment When Stable