---

Source. Published p. 283, Proposition 10.4 (PDF).

Statement. For any field $K$, any $a,b\in K$, and any affine $k$-dimensional subspace $U\subseteq K^n$, at most $2^k$ points of $U$ have all coordinates in $\{a,b\}$.

Proof. Write $U=v+V$, where $V$ is a linear $k$-space. A matrix whose rows form a basis of $V$ has rank $k$, so it has $k$ linearly independent columns. Projection onto those coordinate positions is injective on $V$, and therefore on its affine translate $U$. A two-valued point projects into the set $\{a,b\}^k$, which has at most $2^k$ elements. Injectivity proves the result. This includes $k=0$, arbitrary characteristic, and $a=b$. $\square$

The source's basis of the form $(I\ M)$ first requires a suitable permutation of coordinates; the projection proof states that choice explicitly. Odlyzko's cited result motivates the statement but is not an unproved input to this proof.