Congruences with intervals and subgroups modulo a prime

TL;DR

This paper investigates how almost all residues modulo a prime can be represented as products involving small integers and elements of small multiplicative subgroups, advancing understanding of residue representation in modular arithmetic.

Contribution

It introduces new results on residue representation using small integers and subgroups, building on recent work by Cilleruelo and Garaev.

Findings

Most residues modulo p can be represented as products of small integers and subgroup elements.

The results extend previous bounds on residue representation.

New techniques improve understanding of multiplicative structures modulo primes.

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 $({\mathbb Z}/p{\mathbb Z})^*$. These results are based on some ideas, and their modifications, of a recent work of J. Cilleruelo and M. Z. Garaev (2014),