# Continued fraction expansions of algebraic numbers

## TL;DR

This paper investigates the properties of continued fraction expansions of algebraic numbers, establishing conditions for their independence and symmetry, and applying these results to approximation theorems similar to Roth's theorem.

## Contribution

It introduces new criteria for independence and symmetry in continued fractions of algebraic numbers and applies these to prove a Roth-like approximation theorem.

## Key findings

- Same long sub-words imply identical tails in continued fractions.
- Mirror symmetry in expansions indicates quadratic algebraic numbers.
- Results lead to a Roth-type approximation theorem for algebraic numbers.

## Abstract

In this paper we establish properties of independence for the continued fraction expansions of two algebraic numbers. Roughly speaking, if the continued fraction expansions of two irrational algebraic numbers have the same long sub-word, then the two continued fraction expansions have the same tails. If the two expansions have mirror symmetry long sub-words, then both the two algebraic numbers are quadratic. Applying the above results, we prove a theorem analogous to the Roth's theorem about approximation by algebraic numbers.