Multiplicatively badly approximable numbers and generalised Cantor sets

AbstractLet p be a prime number. The p-adic case of the Mixed Littlewood Conjecture states that liminfq→∞q⋅|q|p⋅‖qα‖=0 for all α∈R. We show that with the additional factor of logqloglogq the statement is false. Indeed, our main result implies that the set of α for which liminfq→∞q⋅logq⋅loglogq⋅|q|p⋅‖qα‖>0 is of full dimension. The result is obtained as an application of a general framework for Cantor sets developed in this paper