4. In a Cartesian coordinate system, a point whose both vertical and horizontal coordinates are integers is called a lattice point. Then, the minimum distance from all lattice points in the plane to the line $y=\frac{3}{4} x+\frac{2}{3}$ is equal to ( ).
A. 0
B. $\frac{2}{15}$
C. $\frac{4}{15}$
D. $\frac{8}{15}$
4. B. The distance $d$ from a lattice point $P\left(x_{0}, y_{0}\right)$ in the plane to the line $y=\frac{3}{4} x+\frac{2}{3}$ is 
$d=\frac{\left|9 x_{0}-12 y_{0}+8\right|}{\sqrt{9^{2}+12^{2}}}=\frac{\left|9 x_{0}-12 y_{0}+8\right|}{15}$. Since $9 x_{0}-12 y_{0}$ is divisible by 3, and 8 leaves a remainder of 2 when divided by 3, we have $\mid 9 x_{0}-12 y_{0}+8 \mid \geqslant 2$, thus $d \geqslant \frac{2}{15}$. The equality holds when and only when $\left(x_{0}, y_{0}\right)=(-2,-1)$, hence the minimum value of $d$ is $\frac{2}{15}$.