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

TL;DR

This paper introduces infinite loops mod n and links their properties to potential counterexamples of the $p$-adic Littlewood Conjecture, providing a new reformulation of the problem.

Contribution

It reformulates the $p$-adic Littlewood Conjecture in terms of infinite loops mod n, establishing a novel connection between these objects and conjecture counterexamples.

Findings

Characterization of counterexamples via infinite loops

Equivalence between existence of infinite loops and conjecture failure

Foundation for further investigation in Part II

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.