[Paper Review] Real quadratic fields with a universal quadratic form of given rank have density zero

Vítězslav Kala, Pavlo Yatsyna+1 more|arXiv (Cornell University)|Feb 23, 2023

Analytic Number Theory Research4 citations

TL;DR

This paper establishes that real quadratic fields admitting a universal quadratic form of a given rank have density zero among all squarefree discriminants D > 0. By linking the rank of such forms to the coefficients of the continued fraction expansion of √D and applying bounds on short vectors in lattices, the authors prove that only a sparse set of fields support low-rank universal forms, with the number of such fields up to X growing slower than any positive power of X.

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.

Motivation & Objective

• To determine the natural density of real quadratic fields that admit a universal quadratic form of a given rank.
• To extend the analysis beyond classical lattices to include mO_H-universal lattices representing all totally positive multiples of a fixed integer m.
• To resolve the conjecture that only finitely many real quadratic fields admit universal forms of small rank, by showing such fields are rare.
• To establish explicit upper bounds on the number of such fields up to a given bound X, showing they form a density-zero set.

Proposed method

• Use the continued fraction expansion of √D to characterize the structure of indecomposable elements in the ring of integers of Q(√D).
• Relate the maximal odd-indexed continued fraction coefficient u_{2i+1} to the minimal rank R of an mO_H-universal lattice via a bound on the number of short vectors.
• Apply a sharp estimate on the number of vectors of a given norm in a Z-lattice (Theorem 3.1) to derive an upper bound B(R,m) on the maximal odd-indexed coefficient.
• Use the bound B(R,m) to restrict the set of D for which an R-rank mO_H-universal lattice can exist, based on the continued fraction structure.
• Combine this with a counting result on periodic continued fractions with bounded coefficients (Corollary 2.12) to bound the number of such D ≤ X.
• Derive asymptotic upper bounds on the number of squarefree D ≤ X admitting such lattices, showing sub-polynomial growth.

Experimental results

Research questions

• RQ1 What is the natural density of real quadratic fields Q(√D) that admit a universal quadratic form of a fixed rank R?
• RQ2 How does the structure of the continued fraction of √D constrain the existence of low-rank universal quadratic lattices over Q(√D)?
• RQ3 Can the number of real quadratic fields with a universal form of rank ≤ R be bounded uniformly in terms of R and the fixed integer m?
• RQ4 To what extent do the coefficients of the continued fraction of √D limit the existence of mO_H-universal lattices of small rank?
• RQ5 What is the growth rate of the number of squarefree D ≤ X for which Q(√D) admits a classical mO_H-universal lattice of rank R?

Key findings

• For any ε > 0, the set of squarefree D > 0 such that R_class(Q(√D)) ≤ D^{1/12−ε} has density zero, with the number of such D ≤ X bounded by O(X^{1−3ε/2} (log X)^{3/2}).
• The number of squarefree D ≤ X for which R(Q(√D)) ≤ D^{1/24−ε} is bounded by O(X^{1−3ε} (log X)^{3/2}), confirming that non-classical universal forms also rarely exist in low rank.
• The existence of an mO_H-universal classical lattice of rank R over Q(√D) implies that the maximal odd-indexed continued fraction coefficient of √D is bounded by B(R,m), a function derived from lattice vector counting.
• The number of squarefree D ≤ X admitting an mO_H-universal classical lattice of rank R is bounded by O(B(R,m)^{3/2} X^{7/8} (log X)^{3/2}) for sufficiently large X.
• The results imply that the ranks of universal forms over real quadratic fields are typically very large, growing faster than any power D^{1/12−ε} for almost all D.
• The method overcomes previous limitations in the literature by using short vector bounds and continued fraction control to establish density-zero results for the first time in this context.