Given $\mathscr{A}\subseteq\mathbb{N}$ an infinite set, consider its $h$-fold sumsets
$$h\mathscr{A}:=\left\{\sum_{i=1}^{h} k_i : k_1,\ldots, k_h\in \mathscr{A} \right\},$$
and let "$\simeq$" be the equivalence relation (in $\mathcal{P}(\mathbb{N})$) given by $\mathscr{A}\simeq \mathscr{B}$ if and only if $\mathscr{A}\triangle \mathscr{B} := (\mathscr{A}\setminus \mathscr{B})\cup (\mathscr{B}\setminus \mathscr{A})$ is finite. Being $\mathrm{d}$ the usual asymptotic density,
Q1. If $h\geqslant 2$, does $\mathrm{d}(h\mathscr{A}) = 1$ imply $h\mathscr{A} \simeq \mathbb{N}$?
This is clearly false for $h=1$, for $\mathscr{A}:= \mathbb{N}\setminus \{n^2: n\in\mathbb{N}\}$ is an easy counterexample. For general $h$ I also think this is false (for if not, the case $h=2$ could be easily used to prove Goldbach's conjecture for suff. large numbers from Vinogradov's results), but I wasn't able to come up with counterexamples. Thus my real question is
How to construct counterexamples to Q1? Furthermore, can we at least guarantee from $\mathrm{d}(h\mathscr{A}) = 1$ that $(h+1)\mathscr{A} \simeq \mathbb{N}$, for all sequences $\mathscr{A}\subseteq \mathbb{N}$?
For the second question, here's a related questionrelated question of mine in M.SE.
