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 $0\notin\mathbb{N}$.



Approximation of sets
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.)


