Upper tail large deviations for arithmetic progressions in a random set

Let X_k denote the number of k-term arithmetic progressions in a random subset of Z/NZ or {1,..., N} where every element is included independently with probability p. We determine the asymptotics of log P(X_k ≥ (1 + δ)EX_k ) (also known as the large deviation rate) where p → 0 with p ≥ N⁻ck 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.