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arXiv · 2304.07399

Densities of integer sets represented by quadratic forms

Pete L. Clark·Paul Pollack·Jeremy Rouse·Katherine Thompson
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Abstract

Let $f(t_1,\ldots,t_n)$ 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.

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Pete L. Clark, Paul Pollack, Jeremy Rouse, Katherine Thompson. 2023-04-14. Densities of integer sets represented by quadratic forms. https://arxiv.org/abs/2304.07399

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