Reformulating the p-adic Littlewood Conjecture in terms of infinite loops mod pk

Abstract

This paper introduces the concept of infinite loops mod n and discusses their properties. In particular, it describes how the continued fraction expansions of infinite loops behave poorly under multiplication by the integer n. Infinite loops are geometric in origin, arising from viewing continued fractions as cutting sequences in the hyperbolic plane, however, they also have a nice description in terms of Diophantine approximation: An infinite loop mod n is any real number which has no semi-convergents divisible by n. The main result of this paper is a reformulation of the p-adic Littlewood Conjecture (pLC) in terms of infinite loops. More explicitly, this paper shows that a real number α is a counterexample to pLC if and only if there is some m∈N such that pℓα is an infinite loop mod pm, for all ℓ∈N∪{0}.