least number of planes intersecting a finite number of points in space, but not intersecting origin.

Question

Let $$\mathbb{R}^*=\mathbb{R}-\{0\}$$ and

$$N=\{0,...,n\}$$ and $$\mathcal{M}=\{ A\subseteq \mathbb{R}^3\times\mathbb{R}^* \mid (\forall\mathbb{x}\in N^3:\mathbb{x}\ne 0)(\exists(\mathbb{a},d)\in A)(\mathbb{a}.\mathbb{x}=d)\}$$

(here $N^3=N\times N\times N$ and $\mathbb{a}.\mathbb{x}$ is inner product)

---

what is $$m:=\min\{|A|:A\in \mathcal{M}\}$$?

---

Do you know any theorem that simplifies finding $m$?

Answer 1

At least you can say that $m\le 3n$ by taking a single value $a=(1,1,1)$ and taking as $d$ all possible sums of triplets in $N^3-\{(0,0,0)\}$.

On the other hand you know that $m \ge n-1$ by taking $x=(k,0,0)$ and noticing that two of these points cannot stay on the same plane.