D. Badziahin

Positive Logarithmic Hausdorff Measures of Exceptional Sets for the $p$-adic and $t$-adic Littlewood Conjectures

We prove that if the exceptional set $E_p$ for the $p$-adic Littlewood conjecture is non-empty, then its logarithmic Hausdorff dimension is at least one. More precisely, whenever $E_p$ is non-empty, it has positive Hausdorff measure with respect to the gauge function $ h(r)=\frac{1}{\log(1/r)}. $ In particular, every n...