A Geometric Interpretation of the $p$-adic Littlewood Conjecture

TL;DR

This paper explores a geometric approach to the $p$-adic Littlewood Conjecture, demonstrating how integer multiplication affects continued fractions through orbifold triangulations, revealing exponential growth in partial quotients.

Contribution

It introduces a novel geometric interpretation of integer multiplication of continued fractions using orbifold triangulations, linking geometric structures to number theoretic properties.

Findings

Integer multiplication of continued fractions can be modeled by changing orbifold triangulations.

Eventually periodic continued fractions exhibit exponential growth in partial quotients under multiplication.

The method provides new insights into the structure of continued fractions and their behavior under integer multiplication.

Abstract

This paper investigates integer multiplication of continued fractions using geometric structures. In particular, this paper shows that integer multiplication of a continued fraction can be represented by replacing one triangulation of an orbifold with another triangulation. This method is used to show that eventually periodic continued fractions have partial quotients which have exponential growth when iteratively multiplied by $n$, for $n$ any fixed, natural number.