Can You Construct a Syndetic Set with an Undefined Density? Let $A \subset \mathbb{N}$. Enumerate $A = \{A_1, A_2,...\}$ such that $A_1 \le A_2 \le ...$. We say that $A$ is syndetic if there exists some $M \geq 0$ such that $A_{i+1} - A_i \le M$ for all $i =1,2,..$ (that is, "the gaps of $A$ are uniformly bounded"). The natural density of $A$, if it exists, is defined to be
$$d(A) = \lim_{N \to \infty} \frac{|A \cap \{1,2,..., N\}|}{N} = \lim_{N \to \infty} \frac{N}{A_N}.$$
It is possible that the limit does not exist. The examples of this phenomenon that I've seen all use the same idea. You need to have a set which first contains a lot of elements of $\{1,2,..N\}$, then misses a lot of $\{N, N+1, ..., N'\}$ then has a lot of $\{N' + 1, ..., N''\}$, etc. A common example is given by
$$A = [2^3, 2^5] \cup [2^7, 2^9] \cup ... \cup [2^{4k-1}, 2^{4k+1}] \cup ...$$
Such examples cannot be syndetic. In the specific given example the problem is that the gaps $[2^{4k+1}, 2^{4k + 5}]$ are not bounded as $k \to \infty$.
So my question is: how can one construct a syndetic set with no natural density (if possible)? Even better: can you construct $A \subset \mathbb{N}$ such that $A$ and $\mathbb{N} \setminus A$ are syndetic and such that $A$ has no density?
Thank you very much in advance.

Take the union of the even integers and a subset of odd integers whose density fluctuates (say between 1/4 and 1/8 of odd numbers, to meet the other conditions).
