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

Abstract

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 non-empty $E_p$ has the cardinality of the continuum. We obtain stronger conclusions for the $t$-adic Littlewood conjecture over a finite field $\mathbb F_q$. For every prime power $q$, non-emptiness of the exceptional set $E_q^{(t)}$ implies that its $1/\log(1/r)$-Hausdorff measure is infinite. Moreover, when $q$ is odd, we refine the recent construction of Lai and Sprang~\cite{LaiSprang2026} and prove that \[ \mathcal H^{h_{A_q}}(E_q^{(t)})=\infty, \] where \[ h_{A_q}(r)=\frac{1}{(\log(1/r))^{A_q}}, \qquad A_q=\frac{q-1}{2}\log_2(q-1). \] In characteristic two, the corresponding conclusion with exponent one remains conditional on the existence of a counterexample.