1
vote
0answers
112 views

is there an analogy between fractals and automorphic forms? [on hold]

Disclaimer: this question is rather vague and thus might not be suitable for this site. Still, as I already asked a similar question on MSE and got no feedback, I finally decided to take the plunge ...
5
votes
0answers
79 views

Isotropy of Apollonian disk-packing

Is there any sense in which the "epsilon-tail" of an Apollonian disk-packing (by which I mean the union of the disks of radius less than epsilon) exhibits more and more statistical isotropy as epsilon ...
11
votes
1answer
250 views

Cantor set intersecting a geometric sequence

I was working on a problem involving finding all points in the intersection of the Cantor set $C$ and the geometric sequence $\{ (2/3)^i \}_{i=1}^\infty$. The only points I have in this intersection ...
-1
votes
1answer
138 views

notable inductive proofs relating to fractals

what are notable/ prominent inductive proofs relating to fractals? the motivation for this question is: fractals are very difficult mathematical objects to work with, and many ...
25
votes
2answers
1k views

Fractal-like structures arising from the action of a group on $\mathbb{Z}^2$

Let $G := \langle a, b, c \rangle < {\rm Sym}(\mathbb{Z}^2)$ be the group generated by the permutation $$ a: \ (m,n) \ \mapsto \ (m-n,m) $$ of order $6$ and the involutions $$ b: \ (m,n) \ ...
4
votes
1answer
129 views

Is there a survey of recent work relating to the Hausdorff dimension of sets defined through some restriction of digits?

I am familiar with the work of Helmut Cajar, but his book is thirty years old and it's clear that there has been substantial progress since then. I have been spending a lot of time looking through ...
3
votes
0answers
81 views

Possible Hausdorff dimension of intersection of Besicovitch-Eggleston like sets

Let $b \geq 2$ be an integer and suppose that $v=(p_0,\cdots,p_{b-1})$ be a probability vector. Let $S_{b,v}$ be the set of real numbers whose $b$-ary expansion has the digit $k$ with relative ...
6
votes
1answer
623 views

Is Gouvêa-Mazur's “Infinite Fern” a fractal?

[EDIT]: Following Qiaochu Yuan's comment, it is better to clarify that I do not know what the right definition of a fractal in the following question should be. But a nice answer might contain such a ...
4
votes
2answers
346 views

Algebraicity of the “outer” boundary of the Mandelbrot set

Let $M$ be the Mandelbrot set and let $\lambda\in M, \mu\in \mathbb C$ be algebraic numbers. Let $t_{\lambda,\mu}$ be defined as $$ t_{\lambda,\mu} = \sup \lbrace t\in \mathbb R\colon \lambda +t\mu ...
2
votes
2answers
325 views

Visualizing the l-adic fractal in the partition function p(n)

This page http://www.aimath.org/news/partition/ and this youtube lecture http://www.youtube.com/watch?v=aj4FozCSg8g speak of a fractal in the values of the partition function p(n). "We prove that ...
11
votes
4answers
749 views

Fourier decay rate of Cantor measures

For $0<\theta<\frac{1}{2}$, denote by $C_\theta$ the Cantor set with dissection ratio $\theta$, i.e. the Cantor set obtained from dissection parttern $(\theta, 1-2\theta,\theta)$. It is known ...