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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Self-similar measures
and intersections of Cantor sets


Authors: Yuval Peres and Boris Solomyak
Journal: Trans. Amer. Math. Soc. 350 (1998), 4065-4087
MSC (1991): Primary 26A46; Secondary 26A30, 28A78, 28A80
MathSciNet review: 1491873
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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 $\nu _\lambda ^{\scriptscriptstyle 1/2}$.


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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
Email: peres@stat.berkeley.edu

Boris Solomyak
Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195
Email: solomyak@math.washington.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-98-02292-2
PII: S 0002-9947(98)02292-2
Keywords: Cantor sets, Hausdorff dimension, self-similar measures
Received by editor(s): September 9, 1996
Additional Notes: The authors were supported in part by NSF grants DMS-9404391 and DMS-9500744.
Article copyright: © Copyright 1998 American Mathematical Society