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 Mathematics , 2014, Abstract: Let $E, F\subset \R^d$ be two self-similar sets. Under mild conditions, we show that $F$ can be $C^1$-embedded into $E$ if and only if it can be affinely embedded into $E$; furthermore if $F$ can not be affinely embedded into $E$, then the Hausdorff dimension of the intersection $E\cap f(F)$ is strictly less than that of $F$ for any $C^1$-diffeomorphism $f$ on $\R^d$. Under certain circumstances, we prove the logarithmic commensurability between the contraction ratios of $E$ and $F$ if $F$ can be affinely embedded into $E$. As an application, we show that $\dim_HE\cap f(F)<\min\{\dim_HE, \dim_HF\}$ when $E$ is any Cantor-$p$ set and $F$ any Cantor-$q$ set, where $p,q\geq 2$ are two integers with $\log p/\log q\not \in \Q$. This is related to a conjecture of Furtenberg about the intersections of Cantor sets.
 M. Pourbarat Mathematics , 2013, Abstract: In the present paper, We introduce a pair of middle Cantor sets namely $(C_\alpha, C_\beta)$ having stable intersection, while the product of their thickness is smaller than one. Furthermore, the arithmetic difference $C_\alpha- \lambda C_\beta$ contains at least one interval for each nonzero number $\lambda$.
 Changhao Chen Mathematics , 2014, Abstract: We show that there exist $(d-1)$ - Ahlfors regular compact sets $E \subset \mathbb{R}^{d}, d\geq 2$ such that for any $t< d-1$, we have $\sup_T \frac{\mathcal{H}^{d-1}(E\cap T)}{w(T)^t}<\infty$ where the supremum is over all tubes $T$ with width $w(T) >0$. This settles a question of T. Orponen. The sets we construct are random Cantor sets, and the method combines geometric and probabilistic estimates on the intersections of these random Cantor sets with affine subspaces.
 Mathematics , 2011, DOI: 10.1088/0951-7715/23/11/005 Abstract: Let $\Gamma_{\beta,N}$ be the $N$-part homogeneous Cantor set with $\beta\in(1/(2N-1),1/N)$. Any string $(j_\ell)_{\ell=1}^\N$ with $j_\ell\in\{0,\pm 1,...,\pm(N-1)\}$ such that $t=\sum_{\ell=1}^\N j_\ell\beta^{\ell-1}(1-\beta)/(N-1)$ is called a code of $t$. Let $\mathcal{U}_{\beta,\pm N}$ be the set of $t\in[-1,1]$ having a unique code, and let $\mathcal{S}_{\beta,\pm N}$ be the set of $t\in\mathcal{U}_{\beta,\pm N}$ which make the intersection $\Gamma_{\beta,N}\cap(\Gamma_{\beta,N}+t)$ a self-similar set. We characterize the set $\mathcal{U}_{\beta,\pm N}$ in a geometrical and algebraical way, and give a sufficient and necessary condition for $t\in\mathcal{S}_{\beta,\pm N}$. Using techniques from beta-expansions, we show that there is a critical point $\beta_c\in(1/(2N-1),1/N)$, which is a transcendental number, such that $\mathcal{U}_{\beta,\pm N}$ has positive Hausdorff dimension if $\beta\in(1/(2N-1),\beta_c)$, and contains countably infinite many elements if $\beta\in(\beta_c,1/N)$. Moreover, there exists a second critical point $\alpha_c=\big[N+1-\sqrt{(N-1)(N+3)}\,\big]/2\in(1/(2N-1),\beta_c)$ such that $\mathcal{S}_{\beta,\pm N}$ has positive Hausdorff dimension if $\beta\in(1/(2N-1),\alpha_c)$, and contains countably infinite many elements if $\beta\in[\alpha_c,1/N)$.
 Mathematics , 2015, Abstract: This paper continues the study of the structure of finite intersections of general multiplicative translates $\mathcal{C}(M_1,\ldots,M_n)=\frac{1}{M_1}\Sigma_{3,\bar{2}}\cap\cdots\cap\frac{1}{M_n}\Sigma_{3,\bar{2}}$ for integers $1\leq M_1<\cdots  M. Pourbarat Mathematics , 2013, Abstract: Suppose that$\mathcal{C}$is the space of all middle Cantor sets. We characterize all triples$(\alpha,~\beta,~\lambda)\in \mathcal{C}\times\mathcal{C}\times \mathbb{R}^*$that satisfy$C_\alpha- \lambda C_\beta=[-\lambda,~1]. $Also all triples (that are dense in$\mathcal{C}\times\mathcal{C}\times \mathbb{R}^*$) has been determined such that$C_\alpha- \lambda C_\beta$forms the attractor of an iterated function system. Then we found a new family of the pair of middle Cantor sets$\mathcal{P}$in a way that for each$(C_\alpha,~ C_\beta)\in\mathcal{P}$, there exists a dense subfield$F\subset \mathbb{R}$such that for each$\mu \in F$, the set$C_\alpha- \mu C_\beta$contains an interval or has zero Lebesgue measure. In sequel, conditions on the functions$f, ~g$and the pair$(C_\alpha,~C_\beta)$is provided which$f(C_{\alpha})- g(C_{\beta})$contains an interval. This leads us to denote another type of stability in the intersection of two Cantor sets. We prove the existence of this stability for regular Cantor sets that have stable intersection and its absence for those which the sum of their Hausdorff dimension is less than one. At the end, special middle Cantor sets$C_\alpha$and$C_\beta$are introduced. Then the iterated function system corresponding to the attractor$C_{\alpha}-\frac{2\alpha}{\beta}C_\beta$is characterized. Some specifications of the attractor has been presented that keep our example as an exception. We also show that$\sqrt{C_{\alpha}}$-$\sqrt{C_{\beta}}\$ contains at least one interval.