The authors compute the dimensions of the sums of Cantor sets. The interest for the support and the dimensions of sums of Cantor sets is a longstanding issue, and has motivated research in geometric measure theory, dynamics and ergodic theory, and number theory. In this nice article, the method is mostly based on ergodic theory.
For \(0<a,b<1/2\), the authors compute the fractal dimension (i.e Hausdorff, correlation, packing) of the measure \(\nu\) obtained by convolution as follows: \(\nu = \mu_a \ast (\mu_b \circ S_\lambda^{-1})\), where \(\mu_a\) is the canonical measure on the central Cantor set \(C_a\) obtained by deleting the central interval of length \(1-2a\) (same for \(\mu_b\)), and \(S_\lambda\) is the dilation \(S_\lambda(x) = \lambda x\). The measure \(\nu\) is supported by \(C_a+ \lambda C_b\). They prove that when the ratio \(\log b / \log a\) is irrational, then for every \(\lambda>0\), the dimension of the measure \(\nu\) is always equal to \(\min( 1, \dim_H(C_a) + \dim_H(C_b))\). In particular, when \(\dim_H(C_a) + \dim_H(C_b) <1\), the dimension of \(\nu\) equals \(\dim_H(C_a) + \dim_H(C_b)\). Hence, according to their vocabulary, there is no resonance between \(C_a\) and \(C_b\), in the sense that there is no gap in the dimension of the sum. A remarkable fact is that it is true for every \(\lambda>0\), not only for almost-every \(\lambda\). The proof is based on two arguments: first a projection argument (\(C_a+\lambda C_b\) is seen as the projection of \(C_a \times C_b\) is some direction), and an ergodic argument enabling them to compute the correlation dimension of the measure \(\nu\).
Finally, it is natural to ask whether the measure \(\nu\) is absolutely continuous with respect to Lebesgue when \(\dim(\nu) =1\) (this question is related to the main questions about Bernoulli convolutions). They show that for \(a=1/4\) and \(b=1/3\), there exists an uncountable set of values \(\lambda\) for which \(\nu\) is singular, attesting once more that there is no easy solutions to this kind of questions.