Tangent-line finite-field Kakeya construction ($K=\{(x,y):x^2-y\text{ is a square}\}\cup \{x=0\}$)

For odd characteristic, the union of the lines $y=mx-m^2/4$ contains a line of every finite slope. For fixed $x$, completing the square shows that it comprises the $(p+1)/2$ values of $y$ for which $x^2-y$ is a square, including zero. Adding the vertical line $x=0$ completes the directions and gives cardinality $(p^2+2p-1)/2=(1/2+o(1))p^2$. This is a finite-field Kakeya set whose density tends to one half.