On the Furstenberg-Katznelson constant for the IP Szemeredi theorem over finite fields

Bergelson et al. observed that Furstenberg's proof of Szemeredi's theorem provides a positive lower bound on the density of arithmetic progressions in sets of positive density in the integers. Namely, for every $\delta\in(0,1]$ and every $k\in \mathbb{N}$, there exists a positive constant $c=c(k,\delta)>0$ such that $$\{n\in \mathbb{N} : d(E\cap (E-n)\cap\dots\cap (E-(k-1)n))>c(k,\delta)\} \neq \emptyset$$ whenever $d(E)\ge \delta$. Similarly, Furstenberg and Katznelson proved the IP Szemeredi theorem, establishing in particular the existence of a constant $c_{\mathrm{IP}}=c_{\mathrm{IP}}(k,\delta)>0$ such that $$\{n\in \mathbb{N} : d(E\cap (E-n)\cap\dots\cap (E-(k-1)n))>c_{\mathrm{IP}}(k,\delta)\}$$ is $\mathrm{IP}^*$ whenever $d(E)\ge \delta$. In this paper, we study analogues of $c$ and $c_{\mathrm{IP}}$ and their ergodic-theoretic counterparts, $c^{\mathrm{rec}}$ and $c_{\mathrm{IP}}^{\mathrm{rec}}$, for vector spaces over finite fields. We provide a qualitative result and in special cases such as Roth's theorem and the IP-Roth theorem, we also provide strong quantitative bounds for these constants. Our tools are primarily ergodic theoretic; we study the characteristic factors and limit of multiple ergodic averages along $\mathrm{IP}$s in vector spaces over finite fields.