Solving quadratic forms in restricted variables with the circle method

Abstract

Let $f(\mathbf{x})$ be a non-singular quadratic form with sufficiently many mixed terms and $t$ an integer. For a sequence of weights $\mathcal{A}$ we study the number of weighted solutions to $f(\mathbf{x})=t$. In particular, we give conditions on both $\mathcal{A}$ and $f$ such that we can use the circle method to count such solutions of bounded height.