Definition: For $A\subset\mathbb{N},$ define the natural density of A to be: $\delta(A)=\lim_{N\rightarrow\infty}\frac{|A\cap\{1,\dots,N\}|}{N}$. We say $\delta(A)$ doesn't exist if the limit $\lim_{N\rightarrow\infty}\frac{|A\cap\{1,\dots,N\}|}{N}$ doesn't exist, i.e. $\liminf_{N\rightarrow\infty}\frac{|A\cap\{1,\dots,N\}|}{N}<\limsup_{N\rightarrow\infty}\frac{|A\cap\{1,\dots,N\}|}{N}$.
Question: Construct an set $A\in\mathbb{N}$ such that $\delta(A)$ exists, but $\delta(A\cap(A+1))$ doesn't exist, where $A+1=\{a+1:a\in A\}$.
My attempt: I firstly considered $B=E \cup O$ where $E=\{\text{even positive integers with an even number of digits (base 10)}\}$, and $O=\{\text{odd positive integers with an odd number of digits (base 10)}\}$. So then $$B=\{1,3,5,7,9,10;12,14,16,\dots,96,98;101,103,\dots,197,199;200 ,202, \dots\}$$ Then $\delta(B)=1/2$. Notice that $$B\cap(B+1)=\{10,101,1000,10001,\dots\}\Rightarrow\delta(B\cap(B+1))=0$$ I want to "perturb" B such that $B\cap(B+1)$ has no natural density.
One of my attempt is that let $B'=E'\cup O$, where $E$ is the set of all integers between $(\frac{10^2k-10^{2k-1}}{2}+10^{2k-1})$ and ${10^{2k}}$, for all $k\in\mathbb{N}$. I.e. let $E'$ be the union of "second halves" of positive integers of an even number of digits. Then $B'\cap (B'+1)$ has no natural density (because the most of the "second halves" are in the intersection, while almost every positive integer of an odd number of digits isn't in the intersection). But the problem is that $\delta(B')$ also does not exist.
I wish I could find a clever "perturbation" of $B$ s.t. $\delta(B)$ exists, but $\delta(B\cap(B+1))$ does not exist.
Could anyone please give me a suggestion or an example that will work?
