Binary and ternary congruences involving intervals and sets modulo a prime

TL;DR

This paper proves that for large primes, certain congruences involving sums of reciprocals with elements from specific sets and intervals can represent any residue, extending known results to binary and ternary cases.

Contribution

It introduces new bounds and methods for representing residues modulo a prime using sums of reciprocals from sets and intervals, including the case when s=1.

Findings

For large primes, any residue can be expressed as a sum of three reciprocals with set elements and bounded denominators.

When s=1, for almost all primes, any residue can be represented as a sum of two such reciprocals.

The results extend previous work on congruences involving intervals and sets modulo primes.

Abstract

Let $s$ be a fixed positive integer constant, $\varepsilon$ be a fixed small positive number. Then, provided that a prime $p$ is large enough, we prove that for any set $\{{\mathcal M}\subseteq \mathbb F_p^*$ of size $|{\mathcal M}|= \lfloor p^{14/29}\rfloor$ and integer $H=\lfloor p^{14/29+\varepsilon}\rfloor$, any integer $\lambda$ can be represented in the form $$ \frac{m_1}{x_1^s}+\frac{m_2}{x_2^s}+\frac{m_3}{x_3^s}\equiv \lambda \bmod p, $$ with $$ m_i\in {\mathcal M}, \quad 1\le x_i\le H, \qquad i=1,2,3. $$ When $s=1$ we show that for almost all primes $p$ the following holds: if $|{\mathcal M}|= \lfloor p^{1/2}\rfloor$ and $H=\lfloor p^{1/2}(\log p)^{6+\varepsilon}\rfloor$, then any integer $\lambda$ can be represented in the form $$ \frac{m_1}{x_1}+\frac{m_2}{x_2}\equiv \lambda \bmod p, $$ with $$ m_i\in {\mathcal M}, \quad 1\le x_i\le H, \qquad i=1,2. $$