Real quadratic fields with a universal quadratic form of given rank have density zero
Vitezslav Kala, Pavlo Yatsyna, Błażej Żmija
TL;DR
The paper investigates when real quadratic fields $\mathbb{Q}(\sqrt{D})$ admit universal quadratic forms of fixed rank over totally real fields. It couples continued fraction analysis of the fundamental unit with explicit short-vector bounds in ${\mathbb Z}$-lattices to translate universality into constraints on lattice representations. The authors prove an explicit upper bound $\mathcal{D}(R,m,X)<A(R,m)\,X^{7/8}(\log X)^{3/2}$ on the count of such fields up to $X$, establishing a density-zero result for real quadratic fields admitting an $m\mathcal{O}_H$-universal lattice of rank $R$. They further show that, for almost all $D$, the minimal ranks grow as powers of $D$ with exponents $1/12-\varepsilon$ (classical) and $1/24-\varepsilon$ (without classical assumption), highlighting that large ranks are typical. The work advances the understanding of universal quadratic forms by providing explicit, quantitative bounds that connect continued fractions, lattice geometry, and universality questions.
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.