High-low method and $p$-adic Furstenberg set over the plane

Abstract: We establish a $p$-adic analogue of a recent significant result of Ren-Wang (arXiv:2308.08819) on Furstenberg sets in the Euclidean plane. Building on the $p$-adic version of the high-low method from Chu (arXiv:2510.20104), we analyze cube-tube incidences in $\mathbb{Q}_p^2$ and prove that for $s < t < 2 - s$, any semi-well-spaced $(s,t)$-Furstenberg set over $\mathbb{Q}_p^2$ has Hausdorff dimension $\ge\frac{3s+t}{2}$. Moreover, as a byproduct of our argument, we obtain the sharp lower bounds $s+t$ (for $0<t\le s\le 1$) and $s+1$ (for $s+t\ge 2$) for general $(s,t)$-Furstenberg sets without the semi-well-spaced assumption, thereby confirming that all three lower bounds match those in the Euclidean case.