A Furstenberg-Katznelson-Weiss type theorem on (d + 1)-point configurations in sets of positive density in finite field geometries

We show that if $E \subset \mathbb{F}_q^d$, the $d$-dimensional vector space over the finite field with $q$ elements, and $|E| \geq ρq^d$, where $q^{-\frac{1}{2}}\ll ρ\leq 1$, then $E$ contains an isometric copy of at least $c ρ^{d-1} q^{d+1 \choose 2}$ distinct $(d+1)$-point configurations.

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