Congruences with intervals and subgroups modulo a prime

Abstract

We obtain new results about the representation of almost all residues modulo a prime $p$ by a product of a small integer and also an element of small multiplicative subgroup of $(\mathbb{Z}/p\mathbb{Z}{)}^{\ast }$. These results are based on some ideas, and their modifications, of a recent work of J. Cilleruelo and M. Z. Garaev (2014),