Digital representations using the greatest integer function

Abstract:

Let ${S_d}(\alpha )$ denote the set of all integers which can be expressed in the form $\sum {{\varepsilon _i}[{\alpha ^i}]}$, with ${\varepsilon _i} \in \{ 0, \ldots ,d - 1\}$, where $d \geq 2$ is an integer and $\alpha \geq 1$ is real, and let ${I_d}$ denote the set of $\alpha$ so that ${S_d}(\alpha ) = {{\mathbf {Z}}^ + }$. We show that ${I_d} = [1,{r_d}) \cup \{ d\}$, where ${r_2} = {13^{1/4}},{r_3} = {22^{1/3}}$ and ${r_2} = {({d^2} - d - 2)^{1/2}}$ for $d \geq 4$. If $\alpha \notin {I_d}$ we show that ${T_d}(\alpha )$, the complement of ${S_d}(\alpha )$, is infinite, and discuss the density of ${T_d}(\alpha )$ when $\alpha < d$. For $d \geq 4$ and a particular quadratic irrational $\beta = \beta (d) < d$, we describe ${T_d}(\beta )$ explicitly and show that $|{T_d}(\beta ) \cap [0,n]|$ is of order ${n^{e(d)}}$, where $e(d) < 1$.