Sums of fractions modulo $p$

TL;DR

This paper investigates the conditions under which elements of a finite field can be expressed as sums of fractions with numerators and denominators from short intervals, advancing understanding of additive properties in finite fields.

Contribution

It introduces new results on the representability of field elements as sums of fractions from short intervals, a problem not extensively studied before.

Findings

Established bounds on the length of intervals needed for representation

Proved that most elements can be expressed as such sums under certain conditions

Extended previous results on additive properties in finite fields

Abstract

Let $\F_p$ be the field of residue classes modulo a large prime $p$. The present paper is devoted to the problem of representability of elements of $\F_p$ as sums of fractions of the form $x/y$ with $x,y$ from short intervals of $\F_p$.