Some arithmetical properties of convergents to algebraic numbers

TL;DR

This paper investigates the intersection of convergents of algebraic irrationals with linear recurrence sequences, showing that an infinite intersection implies the algebraic number is quadratic, and explores related arithmetical properties.

Contribution

It establishes a new criterion linking the intersection of convergents and recurrence sequences to the quadratic nature of algebraic numbers, and analyzes their arithmetical properties.

Findings

Infinite intersection implies the algebraic number is quadratic.

Provides conditions under which convergents and recurrence sequences intersect infinitely.

Discusses arithmetical properties of the sequence of convergents.

Abstract

Let $\xi$ be an irrational algebraic real number and $(p_k / q_k)_{k \ge 1}$ denote the sequence of its convergents. Let $(u_n)_{n \geq 1}$ be a non-degenerate 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 $(u_n)_{n \geq 1}$ is infinite, then $\xi$ is a quadratic number. We also discuss several arithmetical properties of the sequence $(q_k)_{k \ge 1}$.