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

Source 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.