\(\beta\)-expansions and frequency of zero
Let \(\beta> 1\) be a real number, and consider the \(\beta\)-transformation \(T_\beta\) defined on \([0,1)\) by \[ T_\beta x= \beta x- \lfloor\beta x\rfloor. \] Iterations of the map \(T_\beta\) produce expansions of numbers \(x\in [0,1)\) of the form \[ x= \frac {c_1}{\beta}+ \frac {c_2}{\beta^2}+ \cdots\;, \] where \(c_n= c_n(x)= \lfloor\beta T_\beta^{n-1} x\rfloor\). It is well known that \(T_\beta\) is ergodic, and preserves a measure \(\mu_\beta\), known as the Parry measure, which is equivalent to Lebesgue measure on \([0,1)\) (in case \(\beta\) is an integer, then \(\mu_\beta\) equals the normalized Lebesgue measure). By the ergodic theorem one has for a.e. \(x\in [0,1)\), \[ G(\beta)= \mu_\beta ([0,1/\beta))= \lim_{n\to \infty} \frac {\#\{1\leq k\leq n: c_k(x)= 0\}}{n}. \] In this paper the authors study the function \(G(\beta)\) for \(\beta\in (1,\infty)\). They show that 1. \(G(\beta)\geq 1/\beta\) with equality only when \(\beta\) is an integer. 2. \(G(\beta)\) is continuous. 3. If \(\beta\) is simple (i.e. 1 has a finite \(\beta\) expansion), then the right derivative \(G_+'(\beta)= +\infty\).
scientific article; zbMATH DE number 1404419