Spacing and A Large Sieve Type Inequality for Roots of a Cubic Congruence

TL;DR

This paper investigates the distribution of roots of a specific cubic congruence, establishing spacing results and a large sieve inequality, which advances understanding of roots' distribution without proving equidistribution.

Contribution

The authors re-derive a parametrization of roots of a cubic congruence and establish a large sieve inequality for these roots, analogous to quadratic cases, without proving equidistribution.

Findings

Proved spacing properties of roots in the unit square.

Established a large sieve type inequality for roots of the cubic congruence.

Characterized torsion points with expected spacing in the parameter space.

Abstract

Motivated by a desire to understand the distribution of roots of cubic congruences, we re-derive a parametrization of roots $\nu \pmod m$ of $X^3 \equiv 2 \pmod m$ found by Hooley. Although this parametrization does not lead us here to anything towards proving equidistribution of the sequence $\frac{\nu}{m} \pmod 1$, we are able to prove spacing results, and then a large sieve type inequality, which we view as analogous to the large sieve inequality for roots of quadratic congruences found by Fouvry and Iwaniec in their proof that there are infinitely many primes of the form $n^2 + p^2$. The parametrization produces approximations, which are $\asymp m^{2/3}$-torsion points in $\mathbb{R}^2 / \mathbb{Z}^2$ within $O\left(\frac{1}{m} \right)$ of the point $\left( \frac{\nu}{m}, \frac{\nu^2}{m} \right)$. After a digression to characterize those torsion points having the statistically expected spacing, we prove the spacing property alluded to above: that at most a bounded number of the points $\left(\frac{\nu}{m}, \frac{\nu^2}{m}\right)$ with $m \asymp M$ can lie in any disc with radius $\frac{1}{M}$ in $\mathbb{R}^2 / \mathbb{Z}^2$.