arXiv subjects
Publications and source records attributed to M. Pourbarat.
In the present paper, We introduce a pair of middle Cantor sets namely $(C_α, C_β)$ having stable intersection, while the product of their thickness is smaller than one. Furthermore, the arithmetic difference $C_α- λC_β$ contains at least one interval for each nonzero number $λ$.
Suppose that $K$ and $ K'$ are two affine Cantor sets in $\mathbb R$. It is shown that the sum set $K+K'$ has equal box and Hausdorff dimensions, where this common dimension is denoted by $s$, we have $H^s(K+K')<\infty$. It has previously been proven that $H^s(K +K') > 0$, for almost every pair $(K, K')$ satisfying $HD(K)+HD(K') <\frac{1}{2}$ or $HD(K)+HD(K') > 1$. We show that $H^s(K +K') = 0$, for almost every pair $(K, K')$ satisfying the conditions $HD(K) + HD(K') \leq 1$ and $\tau(K) + \tau(K') + 3\tau(K)\tau(K') \geq 1$.