Measure theoretic structure of the sum of two affine Cantor sets

Abstract

Suppose that $K$ and $K^{\prime }$ are two affine Cantor sets in $\mathbb{R}$. It is shown that the sum set $K+K^{\prime }$ has equal box and Hausdorff dimensions, where this common dimension is denoted by $s$, we have $H^s(K+K^{\prime })<\infty$. It has previously been proven that $H^s(K+K^{\prime })>0$, for almost every pair $(K,K^{\prime })$ satisfying $HD(K)+HD(K^{\prime })<\frac{1}{2}$ or $HD(K)+HD(K^{\prime })>1$. We show that $H^s(K+K^{\prime })=0$, for almost every pair $(K,K^{\prime })$ satisfying the conditions $HD(K)+HD(K^{\prime })\le 1$ and $\tau (K)+\tau (K^{\prime })+3\tau (K)\tau (K^{\prime })\ge 1$.

References in corpus (2)

• On the arithmetic sums of Cantor sets
• Resonance between Cantor sets