On distribution of elements of subgroups in arithmetic progressions modulo a prime

Keywords: Let F p be the field of residue classes modulo a large prime number p. We prove that if G is a subgroup of the multiplicative group Fp* and if I? F p is an arithmetic progression, then |G?I|=(1+o(1))|G|I|/p+R, where |R|<(|I|1/2+|G|1/2+|I|1/2|G|3/8p?1/8)po(1). We use this bound to show that the number of solutions to the congruence x n ? ? (mod p), x ? ?, L < x < L + p/n, is at most p 1/3?1/390+o(1) uniformly over positive integers n, ? and L. The proofs are based on results and arguments of Cilleruelo and the author (2014), Murphy, Rudnev, Shkredov and Shteinikov (2017) and Bourgain, Konyagin, Shparlinski and the author

Journal: Proceedings of the Steklov Institute of Mathematics

ISSN: 1531-8605