Integral zeros of quadratic polynomials avoiding sublattices

TL;DR

This paper proves the existence of small-norm integral zeros of quadratic polynomials avoiding certain sublattices, with explicit bounds, and applies these results to the distribution of angles in integer lattices.

Contribution

It establishes explicit bounds for zeros of quadratic polynomials outside sublattices, extending Cassels' theorem and applying to lattice angle distribution.

Findings

Existence of bounded-norm zeros outside finite-index sublattices.

Explicit bounds for these zeros.

Application to the distribution of angles between lattice vectors.

Abstract

Assuming an integral quadratic polynomial with nonsingular quadratic part has a nontrivial zero on an integer lattice outside of a union of finite-index sublattices, we prove that there exists such a zero of bounded norm and provide an explicit bound. This is a contribution related to the celebrated theorem of Cassels on small-height zeros of quadratic forms, which builds on some previous work in this area. We also demonstrate an application of these results to the problem of effective distribution of angles between vectors in the integer lattice.