Some arithmetical properties of convergents to algebraic numbers

Yann Bugeaud and Khoa D. Nguyen

Vol. 326 (2023), No. 1, 17–36

DOI: 10.2140/pjm.2023.326.17

Abstract

Let $\xi$ be an irrational algebraic real number and let ${(p_k∕q_k)}_{k\ge 1}$ denote the sequence of its convergents. Let ${(u_n)}_{n\ge 1}$ be a nondegenerate linear recurrence sequence of integers, which is not a polynomial sequence. We show that if the intersection of the sequences ${(q_k)}_{k\ge 1}$ and ${(u_n)}_{n\ge 1}$ is infinite, then $\xi$ is a quadratic number. This extends an earlier work of Lenstra and Shallit (1993). We also discuss several arithmetical properties of the base-$b$ representation of the integers $q_k$, $k\ge 1$, where $b\ge 2$ is an integer. Finally, when $\xi$ is a (possibly transcendental) non-Liouville number, we prove a result implying the existence of a large prime factor of $q_{k-1}{q}_k{q}_{k+1}$ for large $k$. This is related to earlier results of Erdős and Mahler (1939), Shorey and Stewart (1983), and Shparlinskii (1987).

Keywords

approximation to algebraic numbers, Schmidt subspace theorem, recurrence sequence, continued fraction

Mathematical Subject Classification

Primary: 11J68

Secondary: 11J87

Milestones

Received: 25 October 2022

Revised: 6 January 2023

Accepted: 20 October 2023

Published: 14 December 2023