Multiplicatively badly approximable numbers and generalised Cantor sets

TL;DR

This paper demonstrates that adding a logarithmic factor to a p-adic approximation problem invalidates a conjecture, showing that the set of numbers with certain approximation properties has full dimension, using a new Cantor set framework.

Contribution

It introduces a general framework for constructing Cantor sets and applies it to disprove a variant of the Mixed Littlewood Conjecture with logarithmic factors.

Findings

The set of x with liminf_{q→∞} q log q loglog q |q|_p ||qx|| > 0 has full dimension.

Adding the log q loglog q factor invalidates the conjecture for all real x.

The new Cantor set framework is effective in analyzing multiplicative approximation problems.

Abstract

Let p be a prime number. The p-adic case of the Mixed Littlewood Conjecture states that liminf_{q \to \infty} q . |q|_p . ||q x|| = 0 for all real numbers x. We show that with the additional factor of log q.loglog q the statement is false. Indeed, our main result implies that the set of x for which liminf_{q\to\infty} q . log q . loglog q. |q|_p . ||qx|| > 0 is of full dimension. The result is obtained as an application of a general framework for Cantor sets developed in this paper.