Let $\beta >1$ be a non-integer. We consider $\beta$-expansions of the form ${\sum }_{i=1}^{\infty }{d}_i/{\beta }^i$, where the digits ${\left(d_i\right)}_{i\ge 1}$ are generated by means of a Borel map $K_{\beta }$ defined on ${\{0,1\}}^{\mathbb{N}}\times [0,\lfloor \beta \rfloor /\left(\beta -1\right)]$. We show that $K_{\beta }$ has a unique mixing measure ${\nu }_{\beta }$ of maximal entropy with marginal measure an infinite convolution of Bernoulli measures. Furthermore, under the measure ${\nu }_{\beta }$ the digits ${\left(d_i\right)}_{i\ge 1}$ form a uniform Bernoulli process. In case 1 has a finite greedy expansion with positive coefficients, the measure of maximal entropy is Markov. We also discuss the uniqueness of $\beta$-expansions....

We consider minimal redundant digit expansions in canonical number systems in the gaussian integers. In contrast to the case of rational integers, where the knowledge of the two least significant digits in the “standard” expansion suffices to calculate the least significant digit in a minimal redundant expansion, such a property does not hold in the gaussian numbers : We prove that there exist pairs of numbers whose non-redundant expansions agree arbitrarily well but which have different least significant...

Let E be an interval in the unit interval [0,1). For each x ∈ [0,1) define dₙ(x) ∈ 0,1 by $dₙ\left(x\right):={\sum }_{i=1}^n{1}_E\left(2^{i-1}x\right)\left(mod2\right)$, where t is the fractional part of t. Then x is called a normal number mod 2 with respect to E if $N^{-1}{\sum }_{n=1}^{N}dₙ\left(x\right)$ converges to 1/2. It is shown that for any interval E ≠(1/6, 5/6) a.e. x is a normal number mod 2 with respect to E. For E = (1/6, 5/6) it is proved that $N^{-1}{\sum }_{n=1}^{N}dₙ\left(x\right)$ converges a.e. and the limit equals 1/3 or 2/3 depending on x.