Cho 1 vd s không có mật độ

Đặt $S_k=\{a^k+1,a^k+2,..,a^{k+1}\}, a \in N,a>1$

Chọn $S=\bigcup\limits_{n=0}^{\infty}S_n$
\[→|S_i|=a^{i+1}-a^i, S_i \cap S_j =\phi, \forall i \neq j\]

Vì $|\{x \in S| x \leq a^{2n}\}|=|S_0|+|S_1|+..+|S_{2n-2}|=a^{2n-1}-1$
$|\{x \in S| x \leq a^{2n+1}\}|=|S_0|+|S_1|+..+|S_{2n}|=a^{2n+1}-1$

\[\to \left\{\begin{matrix} \lim_{n \to +\infty}\dfrac{|\{x \in S| x \leq a^{2n}\}|}{a^{2n}}=\lim_{n \to +\infty}\dfrac{a^{2n-1}-1}{a^{2n}}=\dfrac{1}{a} & \\ \lim_{n \to+\infty}\dfrac{|\{x \in S| x \leq a^{2n+1}\}|}{a^{2n+1}}=\lim_{n \to +\infty}\dfrac{a^{2n+1}-1}{a^{2n+1}}=1 & \end{matrix}\right.\]

$→S$ không có mật độ.