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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 $(\log 1/2)/(\log 3)$, 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 $(\log 2)/(\log 3)$, 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).

In the original post I did the calculations incorrectly, and with correct calculations, the original question doesn't make sense. But I'd still like to know where in the literature this sort of connection is spelled out in appropriate generality, with clear definitions of the different notions of fractal dimension that are involved, and perhaps theorems asserting numerical relationships between those quantities. It also now seems to me to be likely that a different sort of definition of fractal dimension than the one I use above would be more appropriate in the discrete setting.

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 $(\log 1/2)/(\log 3)$, 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 $(\log 2)/(\log 3)$, since if you dilate the set by a factor of 3 you get a set that's twice as big (two translates of the Cantor set tile the dilated Cantor set).

Lastly note that $(\log 1/2)/(\log 3) = - (\log 2)/(\log 3)$. 

I'd ike to know where in the literature this sort of connection is spelled out in appropriate generality, with clear definitions of the different notions of fractal dimension that are involved, and theorems asserting numerical relationships between those quantities in a more general setting (applying for instance to the locations of the odd entries in Pascal's triangle and Sierpinski's gasket).

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)?

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 $(\log 1/2)/(\log 3)$, 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 $(\log 2)/(\log 3)$, 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).

In the original post I did the calculations incorrectly, and with correct calculations, the original question doesn't make sense. But I'd still like to know where in the literature this sort of connection is spelled out in appropriate generality, with clear definitions of the different notions of fractal dimension that are involved, and perhaps theorems asserting numerical relationships between those quantities. It also now seems to me to be likely that a different sort of definition of fractal dimension than the one I use above would be more appropriate in the discrete setting.