Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

If a machine did it, it is probably transcendental (even $p$-adically)

Laura Capuano, Sara Checcoli, Marzio Mula, Lea Terracini

Abstract

Continued fraction expansions provide a well-established bridge between algebraic properties of numbers and combinatorics on words. In this article, we investigate the algebraicity of $p$-adic numbers whose continued fractions arise from certain classes of words which generalize the classical automatic, periodic and palindromic words. Our main result shows that, under mild conditions on the $p$-adic continued fraction expansion, such numbers are either algebraic of degree at most 2 or transcendental. This result provides an analogue of results of Bugeaud and Adamczewski-Bugeaud in the real setting and extends previous works that were limited to specific choices of $p$-adic floor functions and less general classes of words.

If a machine did it, it is probably transcendental (even $p$-adically)

Abstract

Continued fraction expansions provide a well-established bridge between algebraic properties of numbers and combinatorics on words. In this article, we investigate the algebraicity of -adic numbers whose continued fractions arise from certain classes of words which generalize the classical automatic, periodic and palindromic words. Our main result shows that, under mild conditions on the -adic continued fraction expansion, such numbers are either algebraic of degree at most 2 or transcendental. This result provides an analogue of results of Bugeaud and Adamczewski-Bugeaud in the real setting and extends previous works that were limited to specific choices of -adic floor functions and less general classes of words.

Paper Structure

This paper contains 23 sections, 21 theorems, 177 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Pokering words: the ♠ and ♣ conditions
  3. Some results on words complexity
  4. Link to automatic words
  5. Link to Sturmian words.
  6. Some non examples
  7. Some more remarks on condition ♠
  8. Some more remarks on condition ♣
  9. Toolbox for the p-adic case
  10. p-adic floor functions
  11. Some useful estimates on the continuants
  12. Estimates for Ruban's and Browkin's floor functions
  13. Periodic and palindromic continued fractions
  14. Our main results and some applications
  15. An explicit version of Theorem \ref{['main-thm-intro']}
  16. ...and 8 more sections

Key Result

Theorem 1.3

Let $\alpha=[0,a_1,a_2,\dots]$, with $a_i\in \mathbb{Z}_{> 0}$, and let $A_n/B_n$ be the $n$-th convergent of its continued fraction expansion. Suppose that the word $a_1 a_2 a_3 \cdots$ satisfies condition $\spadesuit$ or $\clubsuit$, and that there exist a real $C>1$ such that $|B_n|\leq C^n$. The

Figures (1)

  • Figure 1: Families of special words

Theorems & Definitions (41)

  • Definition 1.2
  • Theorem 1.3: Bugeaud
  • Theorem 1.4
  • Corollary 1.5
  • Corollary 1.6
  • Theorem 2.1
  • Proposition 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • ...and 31 more