An inverse theorem on sets with rich additive structure modulo primes

In this paper, we prove several results on the structure of maximal sets $S\subseteq [N]$ such that $S$ mod $p$ is contained in a short arithmetic progression, or the union of short progressions, where $p$ ranges over a subset of primes in an interval $[y,2y]$ with $(\log N{)}^{O(1)}<y\le N$. We also provide several constructions demonstrating the sharpness of our results. Furthermore, as an application, we provide several improvements on the larger sieve bound for $\mathrm{\mid }S\mathrm{\mid }$ when $S$ mod $p$ has strong additive structure, parallel to the work of Green–Harper and Shao for improvements on the large sieve.