The Lebesgue measure of the algebraic difference of two random Cantor sets

Péter Móra, Károly Simon, Boris Solomyak

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Abstract

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.

Original languageEnglish
Pages (from-to)131-149
Number of pages19
JournalIndagationes Mathematicae
Volume20
Issue number1
DOIs
StatePublished - Mar 2009
Externally publishedYes

Bibliographical note

Funding 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 protected] (P. Móra), [email protected] (K. Simon), [email protected] (B. Solomyak).

Funding

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 protected] (P. Móra), [email protected] (K. Simon), [email protected] (B. Solomyak).

FundersFunder number
National Science FoundationDMS-0654408
Directorate for Mathematical and Physical Sciences0654408
Hungarian Scientific Research Fund49835, 71693

    Keywords

    • Branching processes with random environment
    • Difference of Cantor sets
    • Palis conjecture
    • Random fractals

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