# Fractal dimension of a set of points in $\mathbb{R}^2$ [duplicate]

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?

• See here. Nov 29, 2016 at 22:54
• Thanks corey979! I guess this does it numerically. Do you know if the dimension can be found analytically based on the information I provided? Nov 29, 2016 at 23:00
• @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.
– march
Nov 29, 2016 at 23:45
• 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. Nov 29, 2016 at 23:52
• 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) Nov 30, 2016 at 16:52