Real quadratic fields with a universal quadratic form of given rank have density zero
Number Theory
2025-05-23 v3
Abstract
We prove an explicit upper bound on the number of real quadratic fields that admit a universal quadratic form of a given rank, thus establishing a density zero statement. More generally, we obtain such a result for totally positive definite quadratic lattices that represent all the multiples of a given rational integer. Our main tools are short vectors in quadratic lattices combined with an estimate for the number of periodic continued fractions with bounded coefficients.
Keywords
Cite
@article{arxiv.2302.12080,
title = {Real quadratic fields with a universal quadratic form of given rank have density zero},
author = {Vitezslav Kala and Pavlo Yatsyna and Błażej Żmija},
journal= {arXiv preprint arXiv:2302.12080},
year = {2025}
}
Comments
to appear in Amer. J. Math., 18 pages