## Self-similar measures and intersections of Cantor sets

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- by Yuval Peres and Boris Solomyak PDF
- Trans. Amer. Math. Soc.
**350**(1998), 4065-4087 Request permission

## Abstract:

It is natural to expect that the arithmetic sum of two Cantor sets should have positive Lebesgue measure if the sum of their dimensions exceeds 1, but there are many known counterexamples, e.g. when both sets are the middle-$\alpha$ Cantor set and $\alpha \in ({1 \over 3}, \frac 12)$. We show that for*any*compact set $K$ and for a.e. $\alpha \in (0,1)$, the arithmetic sum of $K$ and the middle-$\alpha$ Cantor set does indeed have positive Lebesgue measure when the sum of their Hausdorff dimensions exceeds 1. In this case we also determine the essential supremum, as the translation parameter $t$ varies, of the dimension of the intersection of $K+t$ with the middle-$\alpha$ Cantor set. We also establish a new property of the

*infinite Bernoulli convolutions*$\nu _\lambda ^p$ (the distributions of random series $\sum _{n=0}^\infty \pm \lambda ^n ,$ where the signs are chosen independently with probabilities $(p,1-p)$). Let $1 \leq q_1<q_2 \leq 2$. For $p \neq \frac 12$ near $\frac 12$ and for a.e. $\lambda$ in some nonempty interval, $\nu _\lambda ^p$ is absolutely continuous and its density is in $L^{q_1}$ but not in $L^{q_2}$. We also answer a question of Kahane concerning the Fourier transform of $\nu _\lambda$.

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## Additional Information

**Yuval Peres**- Affiliation: Department of Mathematics, Hebrew University, Jerusalem, Israel
- Address at time of publication: Department of Statistics, University of California, Berkeley, California 94720-3860
- MR Author ID: 137920
- Email: peres@stat.berkeley.edu
**Boris Solomyak**- Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195
- MR Author ID: 209793
- Email: solomyak@math.washington.edu
- Received by editor(s): September 9, 1996
- Additional Notes: The authors were supported in part by NSF grants DMS-9404391 and DMS-9500744.
- © Copyright 1998 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**350**(1998), 4065-4087 - MSC (1991): Primary 26A46; Secondary 26A30, 28A78, 28A80
- DOI: https://doi.org/10.1090/S0002-9947-98-02292-2
- MathSciNet review: 1491873