Shortest Distance in Modular Cubic Polynomials

TL;DR

This paper investigates the minimal size of a box that guarantees at least two points from a modular cubic polynomial, proving that a square of side length roughly p^{1/6} contains such points.

Contribution

It establishes a new bound on the size of a box containing two points of a modular cubic polynomial, advancing understanding of their distribution.

Findings

A square of side length p^{1/6 + ε} contains two points from the polynomial.

Provides bounds on the minimal box size for point pairs in modular cubic polynomials.

Enhances knowledge of the distribution of polynomial points modulo p.

Abstract

In this paper, we study how small a box contains at least two points from a modular cubic polynomial $y \equiv a x^3 + b x^2 + c x + d \pmod p$ with $(a, p) = 1$. We prove that some square of side length $p^{1/6 + \epsilon}$ contains two such points.