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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 $\Pi(E) =\left\{ x \cdot y = x_1y_1+x_2y_2+x_3y_3+x_4y_4: x,y \in E\right\}.$

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

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