Given a subset $S$ of the positive integers $\mathbf{N}$, let $\mathrm{d}^\star(S)$ be its upper asymptotic density, that is, $$ \mathrm{d}^\star(S)=\limsup_{n\to \infty}\frac{|S \cap [1,n]|}{n}. $$ Also, for each integer $k \ge 0$, set $S+k:=\{s+k:s \in S\}$.
Question. Do there exist $X,Y \subseteq \mathbf{N}$ such that $\mathrm{d}^\star(X)>0$, $\mathrm{d}^\star(Y)>0$, and $$ \mathrm{d}^\star(X \cap (Y+k))=0 \,\,\,\text{ for all }k\ge 0? $$
Ps. The answer is positive (constructively, modulo my mistakes) for the following finite-analogue: "Do there exist infinite sets $X,Y \subseteq \mathbf{N}$ such that $X \cap (Y+k)$ is finite for all $k\ge 0$?"
To this aim, let $X$ be the set of squares and $Y$ be the set of cubes which are not also squares. Then, by a result of Siegel, the equation $|x-y|=k$ has finitely many solutions with $x \in X$ and $y \in Y$ for every fixed nonzero integer $k$.
Edit: I just discovered an excellent proof of the finite-analogue without translation invariance in Albiac-Kalton "Topics in Banach Space Theory" (lemma 2.5.3), namely:
"There exists an uncountable family $(A_i)$ of infinite sets of $\mathbf{N}$ such that $A_i \cap A_j$ is finite for all $i\neq j$."
Proof:Identify $\mathbf{N}$ with $\mathbf{Q}$ and, for each irrational number $\theta$ let $(a_{\theta,n})$ be a sequence of rationals converging to $\theta$. Then the uncountable family $$\{\{a_{\theta,n}:n\in \mathbf{N}\}: \theta \in \mathbf{R}\setminus \mathbf{Q}\}$$ satisfies the hypothesis.
