= Sparse peaks can determine a limit superior
{title2=$a_n=\mathbf1_{\{n=2^k\text{ for some }k\geq0\}}$}
Define a <sequence> by $a_n=1$ when $n$ is a power of two, including $1=2^0$, and $a_n=0$ otherwise. Every <tail of a sequence> contains both a later power of two and a later nonpower. Therefore \b[$\liminf a_n=0$ and $\limsup a_n=1$], and the <sequence> does not converge. Its peak indices become increasingly sparse, yet never stop appearing.
Through index $N\geq1$, the number of peaks is $1+\lfloor\log_2N\rfloor$, so their fraction is
$$
\frac{1+\lfloor\log_2N\rfloor}{N}\longrightarrow0.
$$
The <set> of powers of two has <natural density> zero. The same fraction is the <Cesàro mean> of the first $N$ terms, which tends to zero even though the <limit superior> stays one. At $N=1024$, eleven peaks give mean $11/1024\approx0.0107422$. An average and a long-run upper edge answer different questions about the same <sequence>.
\Image[/limsup-sparse-peaks-and-frequency.png]
{title=Peaks of zero <natural density> retain a <limit superior> of one while their <Cesàro means> tend to zero}
{description=Left: the indicator <sequence> has peaks at powers of two; every sufficiently late tail still contains peaks and zeros. Right: the exact peak count divided by $N$ decreases overall toward zero, with jumps when a new power of two enters the count. These finite plots illustrate formulas valid at every positive <integer> index.}
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