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

TL;DR

This paper investigates the behavior of orthogonal transformation-free self-similar sets in three-dimensional space under non-linear projections, showing that certain projections and distance sets have maximal dimension under specific conditions.

Contribution

It demonstrates that projections of these self-similar sets onto the real line have dimension 1 when the set's dimension exceeds 1, and explores the dimension of their distance sets and algebraic products.

Findings

Projections of self-similar sets with dimension > 1 have dimension 1.

Distance sets of such sets have dimension 1.

Third algebraic product of the set has dimension 1 if the set's dimension ≥ 1/3.

Abstract

In the last years considerable attention has been paid for the orthogonal and non-linear projections of self-similar sets. In this paper we consider orthogonal transformation-free self-similar sets in $\mathbb{R}^3$, i.e. the generating IFS has the form $\left\{ \lambda_i \underline{x} + \underline{t}_i \right\}_{i=1}^q$. We show that if the dimension of the set is strictly bigger than $1$ then the projection of the set under some non-linear functions onto the real line has dimension $1$. As an application, we show that the distance set of such self-similar sets 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 $1/3$.