A note on the base-p expansions of putative counterexamples to the p-adic Littlewood conjecture

J. Blackman; S. Kristensen; M. J. Northey

doi:10.1016/j.exmath.2024.125548

A note on the base-p expansions of putative counterexamples to the p-adic Littlewood conjecture

J. Blackman, S. Kristensen^*, M. J. Northey

^*Corresponding author for this work

Department of Mathematics

Research output: Contribution to journal/Conference contribution in journal/Contribution to newspaper › Journal article › Research › peer-review

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→∞|φⁿ(u)|=∞, 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.

Original language	English
Article number	125548
Journal	Expositiones Mathematicae
Volume	42
Issue	3
ISSN	0723-0869
DOIs	https://doi.org/10.1016/j.exmath.2024.125548
Publication status	Published - May 2024

Keywords

Combinatorics on words
Diophantine approximation
p-adic Littlewood conjecture

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10.1016/j.exmath.2024.125548Licence: CC BY

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title = "A note on the base-p expansions of putative counterexamples to the p-adic Littlewood conjecture",

abstract = "In this paper, we investigate the base-p expansions of putative counterexamples to the p-adic Littlewood conjecture of de Mathan and Teuli{\'e}. 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(φω(a)) where φω(a) contains a subword of the form uXuXu with limn→∞|φn(u)|=∞, 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.",

keywords = "Combinatorics on words, Diophantine approximation, p-adic Littlewood conjecture",

author = "J. Blackman and S. Kristensen and Northey, {M. J.}",

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T1 - A note on the base-p expansions of putative counterexamples to the p-adic Littlewood conjecture

AU - Blackman, J.

AU - Kristensen, S.

AU - Northey, M. J.

PY - 2024/5

Y1 - 2024/5

N2 - 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(φω(a)) where φω(a) contains a subword of the form uXuXu with limn→∞|φn(u)|=∞, 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.

AB - 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(φω(a)) where φω(a) contains a subword of the form uXuXu with limn→∞|φn(u)|=∞, 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.

KW - Combinatorics on words

KW - Diophantine approximation

KW - p-adic Littlewood conjecture

U2 - 10.1016/j.exmath.2024.125548

DO - 10.1016/j.exmath.2024.125548

M3 - Journal article

AN - SCOPUS:85189542971

SN - 0723-0869

VL - 42

JO - Expositiones Mathematicae

JF - Expositiones Mathematicae

IS - 3

M1 - 125548

ER -

A note on the base-p expansions of putative counterexamples to the p-adic Littlewood conjecture

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