# $b$-ary expansions of algebraic numbers

## TL;DR

This paper generalizes previous results on the $b$-ary expansions of algebraic numbers, introduces new transcendence criteria, and explores the independence of expansions of algebraic numbers, leading to a generalized Borel conjecture.

## Contribution

It extends earlier work on $b$-ary expansions, providing new transcendence criteria and insights into the independence of algebraic numbers' expansions.

## Key findings

- New transcendence criteria for algebraic numbers
- Linearly independent algebraic numbers have quite independent $b$-ary expansions
- Proposal of a generalized Borel conjecture

## Abstract

In this paper we give a generalization of the main results in \cite{ab,ab1} about $b$-ary expansions of algebraic numbers. As a byproduct we get a large class of new transcendence criteria. One of our corollaries implies that $b$-ary expansions of linearly independent irrational algebraic numbers are quite independent. Motivated by this result, we propose a generalized Borel conjecture.