= Tangent-line finite-field Kakeya construction
{title2=$K=\{(x,y):x^2-y\text{ is a square}\}\cup\{x=0\}$}
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.
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