Construction of uniformly bounded periodic continued fractions

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.