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I have a set of points in $\mathbb{R}^2$, of the form:

$\left(\frac{a}{\ell^2},\frac{b}{\ell^3}\right)$

where $\ell>0$ is an integer and $a$ and $b$ are some real positive numbers.

I am interested to know the fractal dimension of this set of points as $\ell$ becomes infinite. Is there a simple way of computing this?

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  • $\begingroup$ See here. $\endgroup$
    – corey979
    Nov 29, 2016 at 22:54
  • $\begingroup$ Thanks corey979! I guess this does it numerically. Do you know if the dimension can be found analytically based on the information I provided? $\endgroup$
    – user12588
    Nov 29, 2016 at 23:00
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    $\begingroup$ @user12588. Based on your comment (that you are looking for an analytic solution), it looks like you should've posted this on Mathematics rather than here on the Mathematica site. $\endgroup$
    – march
    Nov 29, 2016 at 23:45
  • $\begingroup$ I posted this on Mathematics: the answer is (apparently) obvious. Because the set I presented is countable, its Hausdorff measure is zero. Thanks for all your help anyways. $\endgroup$
    – user12588
    Nov 29, 2016 at 23:52
  • $\begingroup$ The assumption that $a,b$ are "positive real numbers" doesn't really restrict matters unless you are asserting these are fixed for all points in the set. (In which case all these points lie on a common line in the first quadrant) $\endgroup$
    – hardmath
    Nov 30, 2016 at 16:52

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