A generalization of a theorem of Lekkerkerker to Ostrowski's decomposition of natural numbers (Q2880458)

The so-called Zeckendorf expansion of an integer \(n \geq 0\) is the essentially unique way to write \(n = \sum_{2\leq i \leq r} c_i F_i\), where the \(F_i\) are the Fibonacci numbers (with \(F_0 = 0\), \(F_1 = 1\)) and the \(c_i\) obey the rule \(c_i c_{i+1} = 0\) for \(i \geq 2\). \textit{C. G. Lekkerkerker} [Simon Stevin 29, 190--195 (1952; Zbl 0049.03101)] proved that, in the limit, the expected fraction of nonzero coefficients \(c_i\) in the Zeckendorf expansion of \(n\) is \((5-\sqrt{5})/10\).NEWLINENEWLINEIn this rather long, technical, but well-written paper, the authors evaluate the analogous limit for a more general expansion, the Ostrowski expansion (which expresses \(n\) as a linear combination of the denominators of the convergents of a real number \(\alpha\) subject to a certain constraint), in the case where \(\alpha\) is a quadratic irrational.