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

Balázs Bárány¹

¹ Budapest 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

Fundamenta Mathematicae, Tome 237 (2017) no. 1, pp. 83-100.

Résumé

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 $1/3$.

Détail

DOI : 10.4064/fm90-4-2016

Keywords: years considerable attention has paid orthogonal projections non linear images self similar sets paper consider homothetic self similar sets mathbb generating ifs has form lambda underline underline dimension set strictly greater image set under non linear function real line has dimension application distance set self similar set has dimension moreover third algebraic product self similar set itself real line has dimension its dimension least