Hyperplane Furstenberg Sets in Higher Dimensions
The hyperplane Furstenberg problem asks how small the Hausdorff dimension of $E\subset\mathbb{R}^n$ can be when a $t$-dimensional family of affine $(n-1)$-dimensional hyperplanes meets $E$ in sets of dimension at least $s$. For $0<s\le n-1$ and $0<t\le n$, we prove the sharp bound in $\mathbb{R}^n$ $\dim_{\mathrm{H}}E\ge\mathbf{F}(s,t;n,n-1)$, where $\mathbf{F}$ is Gan's Furstenberg index proposed in prime fields. We also construct examples attaining this bound for all parameters. Our main reduction theorem, Theorem 3.1, states that for $n\ge3$, $0<s\le n-2$, and $1<t\le n$, every weighted $(s,t)$ configuration $(E,H)$ at scale $δ$ produces a dual weighted $(s,t-1)$ configuration $(P,\mathcal{T})$ inside a hyperplane $H_{y_0}\cong\mathbb{R}^{n-1}$. The reduction uses radial projection and energy estimates for Frostman measures. Iterating this reduction and applying the Csörnyei--Stull incidence theorem yields the bound.