$p$-adic Sum-Product, Projections, and Furstenberg Sets

Let $p$ be a prime number. We prove the sharp Furstenberg set bound in the $p$-adic plane ${\mathbb{Q}}_p^2$: every $(s,t)$-Furstenberg set $E\subset {\mathbb{Q}}_p^2$ satisfies $${\dim }_{H}E\ge \min \{s+t,\frac{3s+t}{2},s+1\}\mathrm{.}$$ This matches the sharp lower bound in the Euclidean plane. We also derive two related consequences: a $p$-adic projection theorem for the maps ${\pi }_{\theta }(x,y)=x+\theta y$, together with the corresponding exceptional set estimate giving a $p$-adic analogue of Oberlin’s projection question; and a discretized fractal sum-product estimate over ${\mathbb{Q}}_p$, showing that sufficiently non-concentrated subsets of ${\mathbb{Z}}_p^{\times }$ cannot have both small sum set and small product set. The proof follows the projection-theoretic and multiscale machinery developed in the Euclidean works of Orponen-Shmerkin (arXiv:2301.10199) and Ren-Wang (arXiv:2308.08819). The main task is to rebuild this machinery in the non-archimedean setting, and along the way we develop several new $p$-adic inputs needed to overcome the ultrametric features of the problem.