Preprint · arXiv · October 12, 2026

Bounded continued fractions and quadratic orders in $\Q(\sqrt{127})$

Shirali Kadyrov, Aiken Kazin, Alibek Orynbassar, Nurdaulet Shynarbek

Read on arXiv PDFDOI 10.48550/arxiv.2610.16530

Abstract

We study the least periods and multiplier orders in two constructions of Mercat for continued fractions with bounded partial quotients. Our main explicit result concerns a family in $\Q(\sqrt{127})$ with partial quotients at most $4$. For every integer $j\geq1$, we prove that its primitive period has length $92j+2$ and its multiplier order has conductor $y_{4j}/4$, where $(4730624+419775\sqrt{127})^k=x_k+y_k\sqrt{127}$. The totally positive unit group of this order has index $4j$ in that of the maximal order. These exact formulas give divisor-sum lower bounds at fixed traces and a lower bound of order $(\log X)^2$ for the number of bounded continued fractions with discriminant at most $X$. We also prove a primitive-period refinement of Mercat's palindromic construction. Combining this refinement with Hensley's counting theorem gives, for every $m\geq2$ and $η>0$, at least $X^{δ_m/2-η}$ quadratic fields, up to a positive constant, that admit infinitely many purely periodic continued fractions with partial quotients at most $m$. Here $δ_m$ is the Hausdorff dimension of the corresponding continued fraction Cantor set.

Analytic Number Theory ResearchAlgebraic Geometry and Number TheoryMathematical Dynamics and Fractalsquadratic ordersleast periodHausdorff dimension

Bounded continued fractions and quadratic orders in $\Q(\sqrt{127})$

Abstract

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