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

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 $\mathbb{R}^2 / \mathbb{Z}^2$.