TITLE: Much weaker condition for Kakeya sets over finite fields
QUESTION [5 upvotes]: What is the minimum size of a subset $S \subseteq \mathbb{F}_p^n$ such that for all directions $a \in \mathbb{F}_p^n$, there is a line in direction $a$ that intersects $S$ in at least $C$ points?
If $C = p$ this is exactly the finite field Kakeya problem, and we know $|S| \ge C_n p^n$ by the proof of Dvir in On the size of Kakeya sets in finite fields. However, I am curious what bounds we can show if e.g. $C$ is much smaller than $p$, eg. $C = \log^{O(1)}(pn)$.
I believe by random sampling, one can show an upper bound of something like $\lvert S\rvert \le p^{(1-1/C)n}$, which is trivial for $C = \log^{O(1)}(pn)$. A lower bound of $\lvert S\rvert \ge p^{(n-1)/2}$ is also clear, because the points in $S$ determine at most $\lvert S\rvert^2$ lines.
Also, in Dvir's paper, he also handles the case where $C = p^{1-\epsilon}$, but I am asking for even much smaller values of $C$.
So is the truth closer to $p^{n/2}$ or $p^n$?

REPLY [8 votes]: As claimed in equation (2) of this recent paper
Dhar, Manik; Dvir, Zeev; Lund, Ben, Simple proofs for Furstenberg sets over finite fields,  ZBL07471814 [arXiv].
one can get a lower bound of $2^{-n} C^n$ by adapting the arguments from
Bukh, Boris; Chao, Ting-Wei, Sharp density bounds on the finite field Kakeya problem,  ZBL07471810. [arXiv]
For $C=p$ the bound is tight up to a factor of $2$.  I don't know what to conjecture for other values of $C$.
EDIT: for $C=2$ I would imagine by the usual union bound argument that a random set of cardinality $p^{n/2} n^{10} \log^{10} p$ (say) should work to give a upper bound, though I haven't checked this carefully. So maybe the truth is something like $p^{n/2} C^{n/2}$ up to lower order terms?