A new family of expansions of real numbers

Abstract

For $\alpha >1$ we represent a real number in $(0,1]$ in the form \[ \sum_{i=1}^{\infty}(α-1)^{i-1}α^{-(d_{1}+\dots+d_{i})}\] with $d_i\in \mathbb{N}$. We discuss ergodic theoretical and dimension theoretical aspects of this expansion. Furthermore we study their base-change-transformation.