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 $F\subset {\mathbb{R}}^n$, $1\le k\le n-1$, $s\in (0,k]$, 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 $\dim (F\cap (P+a_P))\ge s$. We show that given $k\ge k_0+1$ (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 $$\dim F\ge n-k+s-\frac{k(n-k)-t}{\lceil s\rceil -k_0+1}\mathrm{.}$$ Our methodology is motivated by the work of the second author, Dvir, and Lund over finite fields.