Densities of integer sets represented by quadratic forms

Abstract

Let $f(t_1,\dots ,t_n)$ be a nondegenerate integral quadratic form. We analyze the asymptotic behavior of the function $D_f(X)$, the number of integers of absolute value up to $X$ represented by $f$. When $f$ is isotropic or $n$ is at least $3$, we show that there is a $\delta (f)\in \mathbb{Q}\cap (0,1)$ such that $D_f(X)\sim \delta (f)X$ and call $\delta (f)$ the density of $f$. We consider the inverse problem of which densities arise. Our main technical tool is a Near Hasse Principle: a quadratic form may fail to represent infinitely many integers that it locally represents, but this set of exceptions has density $0$ within the set of locally represented integers.

25 pages