A refinement of Heath-Brown's theorem on quadratic forms

TL;DR

This paper refines Heath-Brown's 1996 circle method approach for counting lattice points on quadratic forms by allowing less smooth weight functions with explicit decay, broadening the method's applicability.

Contribution

It extends Heath-Brown's theorem by relaxing smoothness and vanishing conditions on the weight function, providing a more general and explicit decay framework.

Findings

Allows finitely smooth weight functions that do not vanish at singularities

Provides explicit decay conditions at infinity for the weight function

Uses elementary number theory results, making the method more accessible

Abstract

In his paper from 1996 on quadratic forms Heath-Brown developed a version of the circle method to count points in the intersection of an unbounded quadric with a lattice of short period, if each point is given a weight, and approximated this quantity by the integral of the weight function against a measure on the quadric. The weight function is assumed to be $C_0^\infty$-smooth and vanish near the singularity of the quadric. In our work we allow the weight function to be finitely smooth, not vanish at the singularity and have an explicit decay at infinity. The paper uses only elementary results from the number theory and is available to readers without a number-theoretical background.