Congruences with intervals and arbitrary sets

Given a prime p, an integer H is an element of [1, p), and an arbitrary set M subset of F* p, where F-p is the finite field with p elements, let J(H, M) denote the number of solutions to the congruence xm equivalent to yn mod p for which x, y is an element of [1, H] and m, n is an element of M. In this paper, we bound J(H, M) in terms of p, H, and the cardinality of M. In a wide range of parameters, this bound is optimal. We give two applications of this bound: to new estimates of trilinear character sums and to bilinear sums with Kloosterman sums, complementing some recent results of Kowalski et al. (Stratification and averaging for exponential sums: bilinear forms with generalized Kloosterman sums, 2018, arXiv:1802.09849).