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Using continued fractions with prescribed period for universal quadratic forms

Veronika Mensikova, Helena Muchova

TL;DR

The paper addresses how the residue class of $D$ modulo $n$ can be prescribed when the continued fraction of $\sqrt{D}$ has a given period, employing Friesen’s parametrization to derive congruence constraints that differ by period parity. It then connects these congruence results to universal quadratic forms over $\mathbb{Z}[\sqrt{D}]$, using Kala–Tinková bounds to show that one can construct discriminants in prescribed congruence classes for which universal forms require arbitrarily many variables. This yields existence results for $D$ with specified moduli and period lengths that force large ranks of universal quadratic forms, and highlights open questions about extending these results to even moduli or other period types. Overall, the work strengthens the link between Diophantine properties of $\sqrt{D}$ via continued fractions and arithmetic of quadratic forms over real quadratic fields.

Abstract

We study the congruence classes attained by positive integers $D$ with a prescribed period of the continued fraction of $\sqrt D$. As an application, we refine the available results on large ranks of universal quadratic forms over real quadratic fields by also imposing congruence conditions on their discriminants.

Using continued fractions with prescribed period for universal quadratic forms

TL;DR

The paper addresses how the residue class of modulo can be prescribed when the continued fraction of has a given period, employing Friesen’s parametrization to derive congruence constraints that differ by period parity. It then connects these congruence results to universal quadratic forms over , using Kala–Tinková bounds to show that one can construct discriminants in prescribed congruence classes for which universal forms require arbitrarily many variables. This yields existence results for with specified moduli and period lengths that force large ranks of universal quadratic forms, and highlights open questions about extending these results to even moduli or other period types. Overall, the work strengthens the link between Diophantine properties of via continued fractions and arithmetic of quadratic forms over real quadratic fields.

Abstract

We study the congruence classes attained by positive integers with a prescribed period of the continued fraction of . As an application, we refine the available results on large ranks of universal quadratic forms over real quadratic fields by also imposing congruence conditions on their discriminants.
Paper Structure (4 sections, 11 theorems, 26 equations)

This paper contains 4 sections, 11 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Continued fractions with short period
  3. Continued fractions with general period length
  4. Universal quadratic forms

Key Result

Proposition 2.1

Let $D$ and $t$ be positive integers such that $\sqrt{D} = [t, \overline{2t}]$. Then $D = t^2 + 1$ and $D$ is not divisible by any prime number $p \equiv 3 \pmod{4}$.

Theorems & Definitions (24)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • proof
  • Theorem 2.3
  • proof
  • Theorem 3.1: Friesen friesen1988
  • Theorem 3.2
  • proof
  • Lemma 3.3
  • ...and 14 more