In his paper "On the size of Kakeya sets in finite fields" (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 contining 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 $d=\lfloor q\cdot\min\{\delta,\gamma\}\rfloor-2$.

To what extent does this answer the question?

In his paper "On the size of Kakeya sets in finite fields" (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?

In his paper "On the size of Kakeya sets in finite fields", Dvir introduces the notion of a $(\delta,\gamma)$-Kakeya set, which is a set $K\subset F^n$ such that there exists $L\subset F^n$ of size $|L|\ge\delta q^n$ such that for every $x\in L$ there is a line in the direction $x$ intersecting $K$ in at least $\gamma q$ points. He then proves the following result:

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

To what extent does this answer the question?

In his paper "On the size of Kakeya sets in finite fields" (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 contining 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 $d=\lfloor q\cdot\min\{\delta,\gamma\}\rfloor-2$.

To what extent does this answer the question?