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

In his paper from 1996 on quadratic forms Heath-Brown devel-oped a version of the circle method to count points in the intersection of an unbounded quadric with a lattice of small period, when each point is assigned 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 C0∞-smooth and vanish near the singularity of the quadric. In our work we allow the weight function to be finitely smooth, not to vanish at the singularity and have an explicit decay at infinity. The paper uses only elementary number theory and is available to read-ers with no number-theoretic background. © 2023 Russian Academy of Sciences, Steklov Mathematical Institute of RAS.