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

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 $1/3$.