Mahler's classification and a certain class of $p$-adic numbers

TL;DR

This paper explores the connection between the digit patterns of certain $p$-adic numbers and Mahler's classification, establishing classifications for various digit types and providing an algebraic independence criterion.

Contribution

It establishes a link between digit types of $p$-adic numbers and Mahler's classification, and introduces an algebraic independence criterion for Sturmian digit sequences.

Findings

Automatic, primitive morphic, and Sturmian $p$-adic numbers are classified as $S$-, $T$-, or $U_1$-numbers.

Provides an algebraic independence criterion for Sturmian $p$-adic numbers.

Connects digit pattern complexity with Mahler's classification.

Abstract

In this paper, we study a relation between digits of $p$-adic numbers and Mahler's classification. We show that an irrational $p$-adic number whose digits are automatic, primitive morphic, or Sturmian is an $S$-, $T$-, or $U_1$-number in the sense of Mahler's classification. Furthermore, we give an algebraic independence criterion for $p$-adic numbers whose digits are Sturmian.