= Set without natural density
{title2=$A=\bigcup_{k\geq0}([4^k,2\cdot4^k)\cap\mathbb Z)$}
Include the positive <integers> in $[1,2),[4,8),[16,32),[64,128),\ldots$, and exclude the intervening <integer intervals> and zero. Equivalently, include $[2^{2k},2^{2k+1})\cap\mathbb Z$ for $k\geq0$. Each included block is large enough to raise the running proportion; the following excluded block lowers it again.
With $C_A(N)=|A\cap\{0,\ldots,N-1\}|$, the exact boundary counts follow from a <geometric series>:
$$
C_A(4^k)=\sum_{j=0}^{k-1}4^j=\frac{4^k-1}{3},\qquad C_A(2\cdot4^k)=\sum_{j=0}^k4^j=\frac{4^{k+1}-1}{3}.
$$
The first sum is empty at $k=0$. Dividing by the corresponding boundary, rather than using one denominator for both, gives
$$
\frac{C_A(4^k)}{4^k}=\frac13-\frac{1}{3\cdot4^k},\qquad\frac{C_A(2\cdot4^k)}{2\cdot4^k}=\frac23-\frac{1}{6\cdot4^k}.
$$
For example, $C_A(16)=1+4=5$, whereas $C_A(32)=1+4+16=21$, yielding proportions $5/16$ and $21/32$. For $4^k\leq N\leq2\cdot4^k$, the exact count is $(4^k-1)/3+N-4^k$; for $2\cdot4^k\leq N\leq4^{k+1}$ it stays $(4^{k+1}-1)/3$. Hence the proportion increases through each included block and decreases through each excluded block. These boundary <subsequences> give the actual <limit inferior> and <limit superior>:
$$
\liminf_{N\to\infty}\frac{C_A(N)}N=\frac13,\qquad\limsup_{N\to\infty}\frac{C_A(N)}N=\frac23.
$$
Since these differ, $A$ has no <natural density>. Its <upper asymptotic density> still records its largest limiting proportion.
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