Bounded continued fractions and quadratic orders in Q(√(127))
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(√(127)) with partial quotients at most $4$. For every integer j≥1, we prove that its primitive period has length 92j+2 and its multiplier order has conductor y4j/4, where (4730624+419775√(127))k=xk+yk√(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≥2 and η>0, at least Xδm/2−η quadratic fields, up to a positive constant, that admit infinitely many purely periodic continued fractions with partial quotients at most m. Here δm is the Hausdorff dimension of the corresponding continued fraction Cantor set.
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BibTeX
@misc{kadyrov2026bounded,
title = {{Bounded continued fractions and quadratic orders in $\Q(\sqrt127)$}},
author = {Shirali Kadyrov and Aiken Kazin and Alibek Orynbassar and Nurdaulet Shynarbek},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16530},
eprint = {2610.16530},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16530},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bounded continued fractions and quadratic orders in Q(√(127))
AU - Shirali Kadyrov
AU - Aiken Kazin
AU - Alibek Orynbassar
AU - Nurdaulet Shynarbek
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16530
UR - https://arxiv.org/abs/2610.16530
AB - 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(√(127)) with partial quotients at most $4$. For every integer j≥1, we prove that its primitive period has length 92j+2 and its multiplier order has conductor y_4j/4, where (4730624+419775√(127))^k=x_k+y_k√(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≥2 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.
ER -