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| Palis' Conjecture on sums of Cantor sets claims that typically a sum of two dynamically defined Cantor sets either has zero measure or contains an interval. In 2001 Moreira and Yoccoz proved that generically a sum of two dynamically defined Cantor sets with sums of Hausdorff dimensions larger than one contains an interval, therefore confirming Palis' Conjecture for generic nonlinear Cantor sets. The genericity assumptions there do not allow to apply the result to a specific one- or finite- dimensional family of Cantor sets, which is the setting encountered in many applications. In particular, Palis' Conjecture for affine Cantor sets is still open.
Based on the recent results on convolutions of singular measures (joint with D.Damanik and B.Solomyak), we show that generically (for almost all values of parameters) the sum of two affine Cantor sets has positive Lebesgue measure provided the sum of their Hausdorff dimensions is greater than one (this is a joint result with S.Northrup). Moreover, in the current work in progress (joint with D.Damanik and Y.Takahashi) we show that the sum of a fixed compact set and a Cantor set from a one-parameter family has positive Lebesgue measure for almost all parameters under the assumption that the Hausdorff dimension of the Cantor set changes monotonically with the parameter (once again, provided the sum of Hausdorff dimensions of the compact set and the Cantor set is greater than one). Some applications of the these results to spectral theory will be also given. |
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