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 $\lambda$.

Abstract:
Let $C_\la$ and $C_\ga$ be two affine Cantor sets in $\mathbb{R}$ with similarity dimensions $d_\la$ and $d_\ga$, respectively. We define an analog of the Bandt-Graf condition for self-similar systems and use it to give necessary and sufficient conditions for having $\Ha^{d_\la+d_\ga}(C_\la + C_\ga)>0$ where $C_\la + C_\ga$ denotes the arithmetic sum of the sets. We use this result to analyze the orthogonal projection properties of sets of the form $C_\la \times C_\ga$. We prove that for Lebesgue almost all directions $\theta$ for which the projection is not one-to-one, the projection has zero $(d_\la + d_\ga)$-dimensional Hausdorff measure. We demonstrate the results on the case when $C_\la$ and $C_\ga$ are the middle-$(1-2\la)$ and middle-$(1-2\ga)$ sets.

Abstract:
In this paper we consider some families of random Cantor sets on the line and investigate the question whether the condition that the sum of Hausdorff dimension is larger than one implies the existence of interior points in the difference set of two independent copies. We prove that this is the case for the so called Mandelbrot percolation. On the other hand the same is not always true if we apply a slightly more general construction of random Cantor sets. We also present a complete solution for the deterministic case.

Abstract:
Fractal sets, by definition, are non-differentiable, however their dimension can be continuous, differentiable, and arithmetically manipulable as function of their construction parameters. A new arithmetic for fractal dimension of polyadic Cantor sets is introduced by means of properly defining operators for the addition, subtraction, multiplication, and division. The new operators have the usual properties of the corresponding operations with real numbers. The combination of an infinitesimal change of fractal dimension with these arithmetic operators allows the manipulation of fractal dimension with the tools of calculus.

Abstract:
In this paper, we consider a family of random Cantor sets on the line and consider the question of whether the condition that the sum of the Hausdorff dimensions is larger than one implies the existence of interior points in the difference set of two independent copies. We give a new and complete proof that this is the case for the random Cantor sets introduced by Per Larsson.

Abstract:
Using a slight modification of an argument of Croot, Ruzsa and Schoen we establish a quantitative result on the existence of a dilated copy of any given configuration of integer points in sparse difference sets. More precisely, given any configuration $\{v_1,...,v_\ell\}$ of vectors in $\mathbb{Z}^d$, we show that if $A\subset[1,N]^d$ with $|A|/N^d\geq C N^{-1/\ell}$, then there necessarily exists $r\ne0$ such that $\{rv_1, ...,rv_\ell\}\subseteq A-A$.

Abstract:
We present a complete proof of a theorem of C.G. Moreira. Under mild checkable conditions, the theorem asserts that the Hausdorff dimension of the arithmetic sum of two dynamically defined Cantor subsets of the real line, equals either the sum of the dimensions or 1, whichever is smaller.

Abstract:
Both Cantor middle-third set and Sierpi\'nski carpet are self-similar, perfect, compact metric spaces. In spite of the similarity of the mathematical procedure of construction, there exists between them a fundamental difference in topological nature, and this difference affects the methods of construction of an interesting non-trivial quotient space of them. The totally disconnectedness (or, more generally, zero-dimensional) enables Cantor middle-third set to have a non-trivial quotient space which is self-similar. On the other hand, concerning Sierpi\'nski carpet, because of the connectedness of its structure, no non-trivial quotient space which is self-similar can be constructed by such an elegant procedure as that for Cantor middle-third set. Various topologically significant nature specific to Cantor middle-third set owe mainly to the totally disconnectedness of the set.

Abstract:
Let $C_a$ be the central Cantor set obtained by removing a central interval of length $1-2a$ from the unit interval, and continuing this process inductively on each of the remaining two intervals. We prove that if $\log b/\log a$ is irrational, then \[ \dim(C_a+C_b) = \min(\dim(C_a) + \dim(C_b),1), \] where $\dim$ is Hausdorff dimension. More generally, given two self-similar sets $K,K'$ in $\RR$ and a scaling parameter $s>0$, if the dimension of the arithmetic sum $K+sK'$ is strictly smaller than $\dim(K)+\dim(K') \le 1$ (``geometric resonance''), then there exists $r<1$ such that all contraction ratios of the similitudes defining $K$ and $K'$ are powers of $r$ (``algebraic resonance''). Our method also yields a new result on the projections of planar self-similar sets generated by an iterated function system that includes a scaled irrational rotation.