AI 中文总结
该研究证明p进Littlewood猜想例外集若非空则具正对数Hausdorff测度，还得到有限域上t进情形的更强结论，改进了Lai和Sprang的构造，特征2情形的结论仍待反例存在性验证。
AI 中文摘要
我们证明，若p进Littlewood猜想的例外集$E_p$非空，则其对数Hausdorff维数至少为1。更确切地说，只要$E_p$非空，它关于规范函数$ h(r)=\frac{1}{\log(1/r)} $就具有正的Hausdorff测度。特别地，每个非空的$E_p$都具有连续统的基数。我们在有限域$\mathbb F_q$上针对t进Littlewood猜想得到了更强的结论。对于每个素数幂$q$，例外集$E_q^{(t)}$非空意味着其$1/\log(1/r)$-Hausdorff测度是无穷大。此外，当$q$为奇数时，我们改进了Lai和Sprang近期的构造\cite{LaiSprang2026}，并证明\\[ \mathcal H^{h_{A_q}}(E_q^{(t)})=\infty, \\]其中\\[ h_{A_q}(r)=\frac{1}{(\log(1/r))^{A_q}}, \qquad A_q=\frac{q-1}{2}\log_2(q-1). \\]在特征2的情形下，指数为1的对应结论仍以存在反例为条件。
英文摘要
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.
Comments14 pages
