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

TL;DR

This paper develops a $p$-adic analogue of a recent Euclidean Furstenberg set result, using the high-low method to analyze incidences and establish dimension bounds over $Q_p^2$, confirming Euclidean bounds in the $p$-adic setting.

Contribution

It introduces a $p$-adic version of the high-low method and proves new Hausdorff dimension bounds for $p$-adic Furstenberg sets, extending Euclidean results.

Findings

Established Hausdorff dimension lower bounds for semi-well-spaced $(s,t)$-Furstenberg sets over $Q_p^2$

Derived sharp bounds for general $(s,t)$-Furstenberg sets matching Euclidean cases

Extended the high-low incidence method to the $p$-adic setting.

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.