ON MULTIPLICATIVELY BADLY APPROXIMABLE NUMBERS

The Littlewood conjecture states that $\liminf _{q\to \infty } q\cdot \|q\alpha \|\cdot \|q\beta \|=0$ for all (α,β)∈ℝ2. We show that with the additional factor of log q⋅log log q, the statement is false. Indeed, our main result implies that the set of (α,β) for which $ \liminf _{q\to \infty } q\cdot \log q\cdot \log \log q\cdot \|q\alpha \|\cdot \|q\beta \|>0$ is of full dimension.