Multiplicatively badly approximable numbers and generalised Cantor sets

Let p be a prime number. The p-adic case of the Mixed Littlewood Conjecture states that for all α∈R. We show that with the additional factor of the statement is false. Indeed, our main result implies that the set of α for which is of full dimension. The result is obtained as an application of a general framework for Cantor sets developed in this paper.