Intersections of middle-$\alpha$ Cantor sets with a fixed translation

Abstract

For $\lambda \in (0,1/3]$ let $C_{\lambda }$ be the middle-$(1-2\lambda )$ Cantor set in $\mathbb{R}$. Given $t\in [-1,1]$, excluding the trivial case we show that \[ Λ(t):=\left\{λ\in(0,1/3]: C_λ\cap(C_λ+t)\ne\emptyset\right\} \] is a topological Cantor set with zero Lebesgue measure and full Hausdorff dimension. In particular, we calculate the local dimension of $\Lambda (t)$, which reveals a dimensional variation principle. Furthermore, for any $\beta \in [0,1]$ we show that the level set \[ Λ_β(t):=\left\{λ\inΛ(t): \dim_H(C_λ\cap(C_λ+t))=\dim_P(C_λ\cap(C_λ+t))=β\frac{\log 2}{-\log λ}\right\} \] has equal Hausdorff and packing dimension $(-\beta \log \beta -(1-\beta )\log \frac{1-\beta }{2})/\log 3$. We also show that the set of $\lambda \in \Lambda (t)$ for which ${\dim }_H(C_{\lambda }\cap (C_{\lambda }+t))\ne {\dim }_P(C_{\lambda }\cap (C_{\lambda }+t))$ has full Hausdorff dimension.

32 pages, 3 figures

References in corpus (1)

• Unique expansions and intersections of Cantor sets