Spread Furstenberg Sets
Paige Bright, Manik Dhar
TL;DR
This work advances the Furstenberg set program to high-dimensional flats by introducing spread Furstenberg sets and deriving sharp Hausdorff dimension lower bounds for them in ${\mathbb{R}}^n$. The authors adapt a multi-stage strategy—slice-based reductions (Mattila), Kakeya-type maximal bounds for $k$-flats (Bourgain–Oberlin), and known Furstenberg-type bounds (Héra)—to the spread setting, establishing a dimension lower bound of $\dim F \ge n-k+s-\frac{k(n-k)-t}{\lceil s\rceil-k_0+1}$ for large enough $k$. A warm-up argument with all directions ($t=k(n-k)$) and a full proof using maximal function bounds underpin the main theorem, while a hyperplane analysis via projective transformations and point-hyperplane duality connects spread-Furstenberg results to standard Furstenberg statements. The results extend the Euclidean theory of $(s,t;k)$-Furstenberg sets and align with finite-field methods inspired by Dhar, Dvir, and Lund, with notable corollaries for hyperplane cases. Overall, the paper links geometric measure theory, maximal function bounds, and projective duality to yield new dimension bounds for high-dimensional spread Furstenberg sets, enriching the landscape between Kakeya-style problems and multiscale incidence geometry.
Abstract
We obtain new bounds for (a variant of) the Furstenberg set problem for high dimensional flats over $\mathbb{R}^n$. In particular, let $F\subset \mathbb{R}^n$, $1\leq k \leq n-1$, $s\in (0,k]$, and $t\in (0,k(n-k)]$. We say that $F$ is a $(s,t;k)$-spread Furstenberg set if there exists a $t$-dimensional set of subspaces $\mathcal P \subset \mathcal G(n,k)$ such that for all $P\in \mathcal P$, there exists a translation vector $a_P \in \mathbb{R}^n$ such that $\dim(F\cap (P + a_P)) \geq s$. We show that given $k \geq k_0 +1$ (where $k_0:= k_0(n)$ is sufficiently large) and $s>k_0$, every $(s,t;k)$-spread Furstenberg set $F$ in $\mathbb{R}^n$ satisfies \[ \dim F \geq n-k + s - \frac{k(n-k) - t}{\lceil s\rceil - k_0 +1 }. \] Our methodology is motivated by the work of the second author, Dvir, and Lund over finite fields.