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
>  $$
> \mathrm{d}^\star(X \cap (Y+k))=0 \,\,\,\text{ for all }k\ge 0?
> $$
> 
> 
> 




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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](https://books.google.com/books?id=FkjCulO5xg0C&pg=PA45)), 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.



Density of intersection with shifted sets
The answer is "yes".


Let $X = \bigcup [4^n, 2 \cdot 4^n)$ and $Y = \bigcup [2 \cdot 4^n, 3 \cdot 4^n)$. Then $d^\star(X), d^\star(Y) \geqslant \tfrac{1}{4}$, but $X \cap (Y + k)$ is a finite set for every $k \geqslant 0$.


