Combinatorial and arithmetical properties of linear numeration systems
Let \(m \geq 2\) and let \(\beta_m\) denote the only real root larger than 1 of the equation \(\beta^m = \beta^{m-1} + \dots + 1.\) Let \(b\) denote the equidistribution on the digit set \(\{0,1\}\). In the present paper questions related to the distribution of sums of the form \[ \sum_{n=1}^{\infty} \delta_n \beta^{-n} \] are investigated, where \(\delta_n\) are independent, identically \(b\)-distributed random variables taking values in the set \(\{0,1\}\). The main result concerns the Garsia entropy \(H_{\beta}\) associated to the above sum. \textit{A. Garsia} [Pac. J. Math. 13, 1159-1169 (1963; Zbl 0126.14901)] proved \(H_{\beta} < 1\) for any PV-number \(\beta\). \textit{J. C. Alexander} and \textit{D. Zagier} [J. Lond. Math. Soc. (2) 44, 121-134 (1991; Zbl 0691.58024)] gave a sharp upper bound for \(H_{\beta} < 1\) when \(\beta=\beta_1 = (1+\sqrt{5})/2\). Here this result is generalized for any \(m\). The main theorem states: Let \(e(k,i)\) denote the number of steps in the subtractive Euclidean algorithm applied to the pair \((k,i)\). Let furthermore \[ \kappa_n = \sum_{{0<i<k \atop \gcd(k,i)=1} \atop (k,i)=n} k \log_2 k \] and \[ {\mathcal T}(x) = 1- {1 \over 2} \left( {1-3x \over 1-x} \right)^2 \sum_{n=1}^{\infty} \kappa_n x^n. \] Then \[ H_{\beta_m} = {1 \over \log_2 \beta_m} {\mathcal T}(2^{-m}). \] After proving this theorem the authors derive precise error bounds for the numerical value of the entropy.
scientific article; zbMATH DE number 1912010