Construction of uniformly bounded periodic continued fractions

TL;DR

This paper constructs infinitely many uniformly bounded periodic continued fractions for real quadratic fields, using specific expansion shapes, and links conjectures in number theory to broader implications.

Contribution

It introduces a method to generate bounded periodic continued fractions in quadratic fields and connects Zaremba's conjecture to McMullen's conjecture.

Findings

Existence of infinitely many quadratic fields with bounded periodic continued fractions.

Construction of continued fractions with only 1s and 2s in infinitely many quadratic fields.

A proof that Zaremba's conjecture implies McMullen's conjecture.

Abstract

We build, for real quadratic fields, infinitely many periodic continuous fractions uniformly bounded, with a seemingly better bound than the known ones. We do that using continuous fraction expansions with the same shape as those of real numbers sqrt(n) + n. It allows us to obtain that there exist infinitely many quadratic fields containing infinitely many continuous fraction expansions formed only by integers 1 and 2. We also prove that a conjecture of Zaremba implies a conjecture of McMullen, building periodic continuous fractions from continued fraction expansions of rational numbers.