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A Note on the Base-$p$ Expansions of Putative Counterexamples to the $p$-adic Littlewood Conjecture

John Blackman, Simon Kristensen, Matthew J. Northey

Abstract

In this paper, we investigate the base-$p$ expansions of putative counterexamples to the $p$-adic Littlewood conjecture of de Mathan and Teulié. We show that if a counterexample exists, then so does a counterexample whose base-$p$ expansion is uniformly recurrent. Furthermore, we show that if the base-$p$ expansion of $x$ is a morphic word $τ(φ^ω(a))$ where $φ^ω(a)$ contains a subword of the form $uXuXu$ with $\lim_{n\to\infty}|φ^n(u)|=\infty$, then $x$ satisfies the $p$-adic Littlewood conjecture. In the special case when $p=2$, we show that the conjecture holds for all pure morphic words.

A Note on the Base-$p$ Expansions of Putative Counterexamples to the $p$-adic Littlewood Conjecture

Abstract

In this paper, we investigate the base- expansions of putative counterexamples to the -adic Littlewood conjecture of de Mathan and Teulié. We show that if a counterexample exists, then so does a counterexample whose base- expansion is uniformly recurrent. Furthermore, we show that if the base- expansion of is a morphic word where contains a subword of the form with , then satisfies the -adic Littlewood conjecture. In the special case when , we show that the conjecture holds for all pure morphic words.
Paper Structure (14 sections, 20 theorems, 47 equations)

This paper contains 14 sections, 20 theorems, 47 equations.

Table of Contents

  1. Introduction
  2. Notation
  3. Morphic Words
  4. Main Results
  5. The $p$-adic Littlewood Conjecture
  6. Results on Morphic Words
  7. Applications to the 2-adic Littlewood Conjecture
  8. The $p$-adic Littlewood Conjecture
  9. Proof of Theorems \ref{['Theorem:3_magic_no.']} and \ref{['Theorem:conjugate_pLC']}
  10. Proof of Proposition \ref{['Prop:LimitWord']} and Theorem \ref{['Theorem:unif_rec_CE']}
  11. The $2$-adic Littlewood Conjecture
  12. Proof of Theorem \ref{['Theorem:pure_morphic_2LC']}
  13. Proof of Corollary \ref{['Cor:pure_morphic_2LC']}
  14. Proof of Theorem \ref{['Theorem:bounded_overlap_free_2LC']}

Key Result

Lemma 1.1

(dMT:2004) For each $k\in\mathbb{Z}_{\geq{0}}$, let $\overline{p^kx}=[a_{0,k};a_{1,k},\ldots]$ be the continued fraction expansion of $p^kx$. Then condition (eq:pLC) is equivalent to

Theorems & Definitions (32)

  • Lemma 1.1
  • Definition 1.2
  • Example 1.3
  • Theorem 2.1
  • Corollary 2.2
  • Theorem 2.3
  • Corollary 2.4
  • Proposition 2.5
  • Theorem 2.6
  • Remark 2.7
  • ...and 22 more