The finite-field Furstenberg set lower-bound conjecture

Assume $n\geq 2$, $0\leq\beta\leq 1$, and $p$ is prime. Let $S\subseteq\mathbb{F}_p^n$ be a subset such that for every direction there is a line $l$ in that direction satisfying

$$|l\cap S|\geq p^{\beta}.$$

Finite-field Furstenberg set conjecture. Then

$$|S|\gtrsim q^{\frac{n-1}{2}+\frac{n+1}{2}\beta}.$$

The preceding construction shows that this exponent is attained up to constants in the paper's examples, so the conjecture asserts sharpness for general prime fields. The statement contains the source's notation $q$ although the hypotheses specify the prime field $\mathbb{F}_p$; this should be checked against the original definition of $q$.