# Does there exist a subset $E \in \mathbb{Z}_{p^2}^4$ such that $\Pi(E) \neq \mathbb{Z}_p$?

Denote $$\mathbb{Z}_{p^2}$$ be the ring residues modulo $$p^2,$$ i.e $$\mathbb{Z}_{p^2} = \left\{ 0,1,2,\dots, p^2-1\right\}.$$

$$\mathbb{Z}_{p^2}^{d} = \underbrace{\mathbb{Z}_{p^2} \times \dots \times \mathbb{Z}_{p^2}}_{d \text{ times}}$$

Question: For $$d=4,$$ does there exist a subset $$E \subset \mathbb{Z}_{p^2}^4$$ such that $$|E| > c.p^{7}$$ where $$c$$ is a small positive real constant and $$E$$ satifies $$\Pi(E) \neq \mathbb{Z}_{p^2},$$ where $$\Pi(E) =\left\{ x \cdot y = x_1y_1+x_2y_2+x_3y_3+x_4y_4: x,y \in E\right\}.$$

• If $p$ is even, one can take $E$ consisting of the vectors with even components. – Max Alekseyev Feb 23 at 3:00