Densities of integer sets represented by quadratic forms

被引：0

作者：

Clark, Pete L. ^{[1
]}

Pollack, Paul ^{[1
]}

Rouse, Jeremy ^{[2
]}

Thompson, Katherine ^{[3
]}

机构：

[1] Univ Georgia, Athens, GA 30602 USA

[2] Wake Forest Univ, Winston Salem, NC 27109 USA

[3] US Naval Acad, Annapolis, MD USA

来源：

JOURNAL OF NUMBER THEORY | 2024年 / 256卷

关键词：

Quadratic form; Density; Hasse principle; Inverse problem; LOCAL-DENSITIES; ADC FORMS; BOUNDS;

D O I：

10.1016/j.jnt.2023.09.010

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

Let f (t1, ... , tn) be a nondegenerate integral quadratic form. We analyze the asymptotic behavior of the function Df (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) is an element of Q boolean AND (0, 1) such that Df(X) similar to 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.

引用

页码：290 / 328

页数：39

共 29 条

[1] The integers represented by sets of ternary quadratic forms [J].