In this paper we consider a family of random Cantor sets on the line. We give some sufficient conditions when the Lebesgue measure of the arithmetic difference is positive. Combining this with the main result of a recent joint paper of the second author with M. Dekking we construct random Cantor sets F1, F2 such that the arithmetic difference set F2 - F1 does not contain any intervals but ℒeb(F2 - F1)> 0 almost surely, conditioned on non-extinction.
Bibliographical noteFunding Information:
MSC: Primary 28A80; Secondary 60J80, 60J85 Key words and phrases: Random fractals, Difference of Cantor sets, Palis conjecture, Branching processes with random environment ✩P. Móra was supported by OTKA Foundation #TS 49835. K. Simon was supported by OTKA Foundation #K 71693. B. Solomyak was supported by NSF grant DMS-0654408. E-mails: email@example.com (P. Móra), firstname.lastname@example.org (K. Simon), email@example.com (B. Solomyak).
- Branching processes with random environment
- Difference of Cantor sets
- Palis conjecture
- Random fractals