On homeomorphic product measures on the Cantor set

Multigeometric sequences and Cantorvals

For a sequence $x \in l_1 \setminus c_{00}$, one can consider the achievement set $E(x)$ of all subsums of series $\sum_{n=1}^{\infty} x(n)$. It is known that $E(x)$ is one of the following types of sets: * finite union of closed intervals, * homeomorphic to the Cantor set, * homeomorphic to the set $T$ of subsums of $\sum_{n=1}^{\infty} c(n)$ where $c(2n-1)=\frac{3}{4^n}$ and $c(2n)=\frac{2}{4^n}$ (Cantorval). Based on ideas of Jones and Velleman, and Guthrie and Nymann we describe families of sequences which contain, according to our knowledge, all known examples of $x$'s with $E(x)$ being C…

A pair of non-homeomorphic product measures on the Cantor set

For r in [0,1] let μ_r be the Bernoulli measure on the Cantor set given as the infinite power of the measure on the two-point set with weights r and 1-r. For r and s in [0,1] it is known that the measure μ_r is continuously reducible to μ_s (that is, there is a continuous map sending μ_r to μ_s) if and only if s can be written as a certain kind of polynomial in r; in this case s is said to be binomially reducible to r. In this paper we answer in the negative the following question posed by Mauldin: Is it true that the product measures μ_r and μ_s are homeomorphic if and only if each is a conti…

厚 Cantor 集乘积中的无限常数间隙长度树

Infinite constant gap length trees in products of thick Cantor sets

本文证明充分厚的 Cantor 集乘积可在平面上生成相邻顶点间距为常数的树，且该距离取值集合具有非空内部，并允许树为可数无限，扩展了分形集中的模式研究。

Finite Point configurations in Products of Thick Cantor sets and a Robust Nonlinear Newhouse Gap Lemma

In this paper we prove that the set of tuples of edge lengths in $K_1\times K_2$ corresponding to a finite tree has non-empty interior, where $K_1,K_2\subset \mathbb{R}$ are Cantor sets of thickness $τ(K_1)\cdot τ(K_2) >1$. Our method relies on establishing that the pinned distance set is robust to small perturbations of the pin. In the process, we prove a nonlinear version of the classic Newhouse gap lemma, and show that if $K_1,K_2$ are as above and $ϕ: \mathbb{R}^2\times \mathbb{R}^2 \rightarrow \mathbb{R} $ is a function satisfying some mild assumptions on its derivatives, then there exist…

The Favard length of product Cantor sets

Nazarov, Peres and Volberg proved recently that the Favard length of the $n$-th iteration of the four-corner Cantor set is bounded from above by $n^{-c}$ for an appropriate $c$. We generalize this result to all product Cantor sets whose projection in some direction has positive 1-dimensional measure.

Generalised Cantor sets and the dimension of products

In this paper we consider the relationship between the Assouad and box-counting dimension and how both behave under the operation of taking products. We introduce the notion of `equi-homogeneity' of a set, which requires a uniformity in the size of local covers at all lengths and at all points. We prove that the Assouad and box-counting dimensions coincide for sets that have equal upper and lower box-counting dimensions provided that the set `attains' these dimensions (analogous to `s-sets' when considering the Hausdorff dimension), and the set is equi-homogeneous. Using this fact we show that…