Lemma 4.2. Assuming that:

$A \subseteq 𝔽_{3}^{n}$
$P \in V_{n}^{d}$ is a polynomial
$P (a + a^{'}) = 0$ for all $a \neq a^{'} \in A$

Then

| {a \in A : P (2 a) \neq 0} | \leq 2 m_{d ∕ 2} .