Measure theoretic structure of the sum of two affine Cantor sets

M. Pourbarat

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arXiv · 1306.5880

Measure theoretic structure of the sum of two affine Cantor sets

M. Pourbarat

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Abstract

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 $\tau(K) + \tau(K') + 3\tau(K)\tau(K') \geq 1$.

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M. Pourbarat. 2013-06-25. Measure theoretic structure of the sum of two affine Cantor sets. https://arxiv.org/abs/1306.5880

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