Densities of integer sets represented by quadratic forms

2023/04/14 by Clark, Pete L., Pollack, Paul, Rouse, Jeremy +1
#FOS: Mathematics #Number Theory (math.NT) #Primary 11E12 #Secondary 11E20

paper · doi:10.48550/arxiv.2304.07399

Abstract

Let f(t₁,…,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 δ(f) ∈ ℚ ∩ (0,1) such that D_f(X) ∼ δ(f) X and call δ(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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