The classical finite field Kakeya conjecture state as following (for conveinent, all version of kakeya conjecture is state in hasdorff dimension version):   

$\mathrm{Finite\ Field\ Kakeya\ Conjecture}$: Let $\mathrm{F}=\mathrm{F}/\mathrm{qF}$ be a finite field, let $K \subseteq \mathrm{F}^{\mathrm{n}}$ be a Kakeya set, i.e. for each vector $y \in \mathbf{F}^{n}$ there exists $x \in \mathbf{F}^{n}$ such that $K$ contains a line $\{x+t y: t \in \mathbf{F}\}$. Then the set $K$ has size at least $c\_{n}q^{n}$ where $c\_{n}>0$ is a constant that only depends on $n$.


.....................(\*)


Dvir's proof of the finite field version of Kakeya conjecture is elegant and influential.
On the other hand, it seems the corresponding version of (\*) in $\mathrm{R^n}$ is that,


a set $K$ if satisfied, for every $\theta \in S^{n-1}$ $K$ contain a line $l\_{\theta, a\_{\theta}}$ and $l$ has direction $\theta$ , then the haussdorff dimension of $K$ is $n$.


But this version in Euclidean space is not essential difficult, in particuler because every line in $R^n$ is not compact and a suitable infinite sequence of line shall group a set which has hausdroff dimension $n$. The original version of Kakeya conjecture in $R^n$ change line to a segement with length 1. So, is there a possibility that a more subtle version of finite field analoge of Kakeya conjecture maybe true? i.e.:


$\mathrm{Version \ 1}$: Fix a sequence $\{a\_m\}\_{m=1}^{\infty}$, $a\_{m} \to \infty$ as $m \to \infty$ and we define a set $K$ in $F\_{p\_m}^n= (Z/Z\_{p\_m})^n $ to be a Kakeya set iff for every vector $v\in F\_{p\_m}^n, v\neq 0 $, such that $K$ contain consecutively(or not) at least $a\_n$ point in a line $\{x+t v: t \in \mathbf{F}\}$ for some $x\in F\_{p\_m}^n$, i.e. contain every direction line with
at least $a\_m$ elements, then exist $c>0$ only depend on $m$ such that $|K|\geq cp\_m^{n-1}$?\


or more easier problem(compare with version 1) but the directly use of polynominal method
Invalidation is a following:


$\mathrm{Version \ 2}$: we define a set $K$ in $F\_{p\_m}^n= (Z/Z\_{p\_m})^n $ to be a Kakeya set iff for every vector $v\in F\_{p\_m}^n, v\neq 0 $, such that $K$ contain consecutive(or not) at least $p\_m^{\frac{1}{2}}$ point in a line $\{x+t v: t \in \mathbf{F}\}$ for some $x\in F\_{p\_m}^n$, i.e. contain every direction line with
at least $p\_m^{\frac{1}{2}}$ elements, then exist $c>0$ only depend on $n$ such that $|K|\geq cp\_m^{n-1}$?


for $\mathrm{Version \ 2}$, the polynomial argument gain a bound $O(p^{\frac{m}{2}})$ which coincide with the combinatorics argument two point depending a line. And it is esay to show if take $a\_m=2, \forall m \in N^\*$ in $\mathrm{Version \ 1}$, then this set $K $ in general do not have full dimension, this corresponde to the situation in $R^n$ when we have a scale $\delta$ kakeya set but instead of $T^{\delta}\_{a\_{\theta},\theta}$ is a $1\times \delta^{n-1}$ tube, we only have a $O(\delta) \times\delta^{n-1}$ tube.



Is finite field version kakeya conjecture still true when changing the line of every direction with only 2(or several but not the full line)element?
In his paper "[On the size of Kakeya sets in finite fields](https://arxiv.org/pdf/0803.2336.pdf)" (where the proof of the finite field Kakeya conjecture has appeared), Dvir also introduces the notion of a $(\delta,\gamma)$-Kakeya set, which is, essentially, a set $K\subset\mathbb F\_q^n$ with the following property: there are at least $\delta(q^n-1)/(q-1)$ directions in $\mathbb F\_q^n$ such that in each of these directions there is a line containing at least $\gamma q$ points of $K$. He then proves the following result:



> 
> Theorem. If $K\subset\mathbb F\_q^n$ is a $(\gamma,\delta)$-Kakeya set, then
> $$ |K| \ge \binom{n+d-1}{n-1}, $$
> where $d=\lfloor q\cdot\min\{\delta,\gamma\}\rfloor-2$.
> 
> 
> 


Applying this theorem with $\delta=1$ and $\gamma=\varepsilon q^{-1+c}$, we conclude that if $K\subset\mathbb F\_q^n$ contains at least $\varepsilon q^c$ collinear points in every direction, then
$$ |K| \ge \binom{n+d-1}{n-1},\quad d=\lfloor \varepsilon q^c\rfloor-2. $$


Does this answer the question?


