Intereting Posts

Formula for computing integrals
Is it possible to generalize Ramanujan's lower bound for factorials that he used in his proof of Bertrand's Postulate?
Trick with 3-digit numbers, always get 1089
Convergence of $\sum_{n=1}^{\infty} \log\left(\frac{(2n)^2}{(2n+1)(2n-1)}\right)$
A variation of Borel Cantelli Lemma
We define a sequence of rational numbers {$a_n$} by putting $a_1=3$ and $a_{n+1}=4-\frac{2}{a_n}$ for all natural numbers. Put $\alpha = 2 + \sqrt{2}$
How can I find the number of the shortest paths between two points on a 2D lattice grid?
Finding the homology group of $H_n (X,A)$ when $A$ is a finite set of points
$\lim\limits_{x\to\infty}f(x)^{1/x}$ where $f(x)=\sum\limits_{k=0}^{\infty}\cfrac{x^{a_k}}{a_k!}$.
How to Classify $2$-Plane Bundles over $S^2$?
Find the equation of the tangent line to $y=x^4-4x^3-5x+7$
Most wanted reproducible results in computational algebra
What is the exact definition of a reflexive relation?
Why are these estimates to the German tank problem different?
In how many ways can the letters of the word CHROMATIC be arranged?

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

$\Bigg\{\left(\frac{a}{\ell^2},\frac{b}{\ell^3}\right): \ell \in \mathbb{N}^+\Bigg\}$

where $a$ and $b$ are some real positive numbers.

- Properties of the Mandelbrot set, accessible without knowledge of topology?
- Fractal derivative of complex order and beyond
- Can monsters of real analysis be tamed in this way?
- This one weird trick integrates fractals. But does it deliver the correct results?
- Zoom out fractals? (A question about selfsimilarity)
- Dimension of a Two-Scale Cantor Set

I am interested to know the box dimension of this set. Is there a simple way to determine this analytically?

- How to prove Mandelbrot set is simply connected?
- Why does the mandelbrot set and its different variants follow similar patterns to epi/hypo trochodis and circular multiplication tables?
- Mandelbrot set: periodicity of secondary and subsequent bulbs as multiples of their parent bulbs
- What is the algorithm hiding beneath the complexity in this paper?
- Why does the Hilbert curve fill the whole square?
- Odd and even numbers in Pascal's triangle-Sierpinski's triangle
- Can a fractal be a manifold? if so: will its boundary (if exists) be strictly one dimension lower?
- Sets of Constant Irrationality Measure
- Symmetric Icon Fractals
- Continuous coloring of a Mandelbrot fractal

I believe the answer is $1/3$, independent of the values of $a$ and $b$.

To see this, we first show that the box counting dimension of the set

$$E = \{1,1/4,1/9,\ldots,1/n^2,\ldots\}$$

is $1/3$. The computations below very much mimic those of example 3.5 on page 45 of Falconer’s *Fractal Geometry*.

To this end, let $\varepsilon>0$ and choose $N$ to be the unique natural number such that

$$\frac{1}{N^2} – \frac{1}{(N+1)^2}<\varepsilon\leq\frac{1}{(N-1)^2}-\frac{1}{N^2}.$$

Now, since $\varepsilon \leq 1/(N-1)^2-1/N^2$, we need at least $N-1$ sets of diameter $\varepsilon$ to cover $E$ – one for each of the numbers $1,1/4,\ldots,1/(N-1)^2$. Thus, $N_{\varepsilon}(E) \geq N-1$. And since

$$\varepsilon > \frac{1}{N^2} – \frac{1}{(N+1)^2} > \frac{1}{(N+1)^3},$$

$1/\varepsilon<(N+1)^3$. Thus,

$$\frac{\log(N_{\varepsilon}(E))}{\log(1/\varepsilon)} >

\frac{\log(N-1)}{\log((N+1)^3)} \to \frac{1}{3}.$$

This much shows that

$$\liminf_{\varepsilon\to0^+}\frac{\log(N_{\varepsilon}(E))}{\log(1/\varepsilon)} \geq \frac{1}{3}.$$

To show that

$$\limsup_{\varepsilon\to0^+}\frac{\log(N_{\varepsilon}(E))}{\log(1/\varepsilon)} \leq \frac{1}{3},$$

we use the fact that, since $\varepsilon>1/(N+1)^3$, all the points of $E$ in $[0,1/(N+1)^2]$ may be covered with $N+1$ sets of size $\varepsilon$. That leaves us with only $N$ more points so that $E$ may be covered with $2N+1$ sets of size $\varepsilon$ so that $N_{\varepsilon}(E)\geq 2N+1$. Also using the fact that

$$\varepsilon\leq\frac{1}{(N-1)^2}-\frac{1}{N^2}< \frac{6}{N^3}$$

for $n>2$, we get that

$$\frac{\log(N_{\varepsilon}(E))}{\log(1/\varepsilon)} <

\frac{\log(2N+1)}{\log((N+1)^3/6)} \to \frac{1}{3}.$$

Now, it’s just a couple of steps to get from here to your set of interest – one small step and one not so small.

First, if we define

$$E_a = \{a,a/4,a/9,\ldots,a/n^2,\ldots\}$$

for a positive number $a$, then we again have a set of box dimension $1/3$ because box dimension is preserved under similarity transformations.

Now, let’s denote your set by $E_{a,b}$. The projection of your set onto the $x$-axis does not increase distance. Thus, it cannot increase box dimension so that the lower box dimension of $E_{a,b}$ is at least $1/3$. On the other hand, $E_{a,b}$ is the image of $E_a$ under the function $F:\mathbb R\to\mathbb R^2$ defined by $F(x) = (x,b(x/a)^{3/2})$. Since we’re mapping to the graph of a differentiable function, this is a Lipschitz map and, therefore, also does not increase dimension. So the upper box dimension of $E_{a,b}$ is at most $1/3$. Taking these together, we a well defined box dimension of $1/3$.

- Connected sum of surfaces with boundary
- Let $g_n= {2^2}^n +1 $. Prove $g_0 · g_1 · · · g_{n−1} = g_{n} − 2$
- Finitely presented Group with less relations than Generators.
- Prove that $a+b$ is a perfect square
- Finding the remainder of $\overbrace{11\ldots1}^{123 \text{ times}}$ divided by $271$
- Looking for Open Source Math Software with Poor Documentation
- What does it mean to integrate with respect to the distribution function?
- A question about the definition of a neighborhood in topology
- A diagram which is not the torus
- The final state of 1000 light bulbs switched on/off by 1000 people passing by
- Does the open mapping theorem imply the Baire category theorem?
- How to find all groups that have exactly 3 subgroups?
- How to find center of a conic section from the equation?
- Proving facts about groups with representation theory.
- For a Planar Graph, is it always possible to construct a set of cycle basis, with each and every edge Is shared by at most 2 cycle bases?