Approximation of sets

Question

Is the following true? For every $\varepsilon>0$ there is a finite subset $W$ of $\mathbb{N}\times \mathbb{N}\times \mathbb{N}$, such that $$|p_1(W)\cap p_2(W)\cap \{p_1(x)+p_2(x):x\in W\}\cap \{p_2(x)+p_3(x):x\in W\}\cap \{p_1(x)+p_2(x)+p_3(x):x\in W\}|\geq (1-\varepsilon)|W|.$$

Here $p_i$ is the projection on the $i$-th coordinate of $W$, and $0\notin\mathbb{N}$.

Answer 1

Yes, one can for instance take

$$ W := \{ ( 2^i 3^j, 2^i 3^j, 2^i 3^j): 0 \leq i,j \leq N \}$$

for some large $N$. (There is also the degenerate example in which $W$ is taken to be the empty set, but presumably you wish to exclude this case.)