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On some non-linear projections of self-similar sets in $\mathbb {R}^3$

Volume 237 / 2017

Balázs Bárány Fundamenta Mathematicae 237 (2017), 83-100 MSC: Primary 28A80; Secondary 28A78, 37C45. DOI: 10.4064/fm90-4-2016 Published online: 9 November 2016

Abstract

In the last years considerable attention has been paid to orthogonal projections and non-linear images of self-similar sets. In this paper we consider homothetic self-similar sets in $\mathbb {R}^3$, i.e. the generating IFS has the form $\{\lambda _i\underline {x} +\underline {t} _i\}_{i=1}^q$. We show that if the dimension of the set is strictly greater than $1$ then the image of the set under some non-linear function to the real line has dimension $1$. As an application, we show that the distance set of such a self-similar set has dimension $1$. Moreover, the third algebraic product of a self-similar set with itself on the real line has dimension $1$ if its dimension is at least $1/3$.

Authors

  • Balázs BárányBudapest University of Technology and Economics
    BME-MTA Stochastics Research Group
    P.O. Box 91
    1521 Budapest, Hungary
    and
    Mathematics Institute
    University of Warwick
    Coventry CV4 7AL, UK
    e-mail

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