Spread Furstenberg Sets

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.