= Rich line covering bound over a finite field
{title2=$|N|\ge\binom{n+m-1}{n}$}
Suppose every point of $\mathbb F_q^n$ lies on an <affine line in a vector space> meeting a subset $N$ in at least $m$ points, with $1\le m\le q$. A smaller $N$ would admit a nonzero <multivariate polynomial> of <total degree> at most $m-1$ vanishing on it, by the <dimension of a bounded-total-degree polynomial space>. Each selected line then has more <roots of a polynomial> than the degree of its restriction, so the <polynomial> vanishes at every point. The <Schwartz-Zippel lemma> forbids this because $m-1<q$. Thus the displayed bound holds, and is at least $m^n/n!$. This is a point-covering condition; the <finite-field Kakeya set> condition instead quantifies over directions.
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