Let $S$ be the set of positive integers whose base-three expansion contains only the digits 0 and 2. The discrete set $S$ in a sense has (negative) fractal dimension $\log 2/3$, since if you dilate the set by a factor of 3 you get a set that's half as big (two translates of the dilated set tile the original set).

Meanwhile, the Cantor set (which can be constructed as the scaling limit of initial segments of $S$) has fractal dimension $\log 3/2 = - \log 2/3$, since if you dilate the set by a factor of 3 you get a set that's twice as big (two translates of the original set tile the dilated set).

Is there a place in the literature where this sort of connection is spelled out in appropriate generality, with clear definitions of the different notions of fractal dimension that are involved, and a theorem asserting that numerical relationship we see for the Cantor set holds much more generally (e.g., for the Sierpinski gasket)?