p进数和积、投影与富斯滕贝格集
$p$-adic Sum-Product, Projections, and Furstenberg Sets
AI总结：
研究p进平面\(\mathbb{Q}_p^2\)中富斯滕贝格集的界，通过遵循欧几里得相关工作的投影理论和多尺度机制进行证明，得出p进投影定理、例外集估计及离散分形和积估计等结果。
AI中文摘要：
设p为素数。我们证明了p进平面\(\mathbb{Q}_p^2\)中富斯滕贝格集的精确界：每个\((s,t)\) - 富斯滕贝格集\(E\subset\mathbb{Q}_p^2\)满足\(\dim_H E\ge \min\left\{s+t,\frac{3s+t}{2},s+1\right\}\)。这与欧几里得平面中的精确下界相匹配。我们还得出两个相关结果：映射\(\pi_\theta(x,y)=x+\theta y\)的p进投影定理，以及相应的例外集估计，给出了奥伯林投影问题的p进类似物；以及\(\mathbb{Q}_p\)上的离散分形和积估计，表明\(\mathbb{Z}_p^\times\)中充分非集中的子集不能同时具有小和集与小积集。证明遵循奥波宁 - 施默金（arXiv:2301.10199）和任 - 王（arXiv:2308.08819）在欧几里得工作中发展的投影理论和多尺度机制。主要任务是在非阿基米德环境中重建此机制，在此过程中我们开发了几个新的p进输入来克服问题的超度量特征。
英文摘要：
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 $$ \dim_H E\ge \min\left\{s+t,\frac{3s+t}{2},s+1\right\}. $$ This matches the sharp lower bound in the Euclidean plane. We also derive two related consequences: a $p$-adic projection theorem for the maps $π_θ(x,y)=x+θy$, 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.
