* A question of Erd\H{o}s on whether the set of perfect squares can be close to a sumset, and a multiplicative analogue by Hajdu and S\'{a}rk\"{o}zy. * A conjecture of Van Lint and MacWilliams on the characterization of maximum subsets of a finite field of square order such that pairwise differences are all squares (also known as the Erd\H{o}s-Ko-Rado theorem for Paley graphs), and its generalization. * Inverse sieve problems (that have been studied by Green--Harper, Helfgott--Venkatesh, Shao, and Walsh), motivated by the inverse Goldbach problem.

Joint work with Ernie Croot and Junzhe Mao.