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arXiv · 2101.05112

Infinite Loops and the $p$-adic Littlewood Conjecture. Part I: Reformulating the $p$-adic Littlewood Conjecture in Terms of Infinite Loops

John Blackman
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Abstract

In this paper we introduce the concept of an infinite loop mod $n$ and discuss the properties that these objects have. In particular, we show that a real number $\alpha$ is a counterexample to the $p$-adic Littlewood Conjecture if and only if there exists some $m\in\mathbb{N}$ such that $p^k\alpha$ is an infinite loop mod $p^m$, for all $k\in\mathbb{N}$. This paper is the first of a two part series, which investigate the link between infinite loops and the $p$-adic Littlewood Conjecture.

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John Blackman. 2021-01-13. Infinite Loops and the $p$-adic Littlewood Conjecture. Part I: Reformulating the $p$-adic Littlewood Conjecture in Terms of Infinite Loops. https://arxiv.org/abs/2101.05112

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