Bemerkungen zur Theorie der Diophantischen Approximationen

Initial powers of Sturmian sequences

“…where the sequence of integer digits (c k ) k≥1 satisfies the admissibility condition (1), is called an Ostrowski expansion following [35] (see also [10,18,19,28,29,26,39,40]). Note that the characteristic sequence of slope α corresponds to intercept x = 0, having all c k equal to 0.…”

“…The sequence (a k ) k≥1 turns out to be the sequence of partial quotients of the slope (defined as the density of the symbol 1), while (c k ) k≥1 is the sequence of digits in the arithmetic Ostrowski expansion of the intercept of the Sturmian sequence (see for instance [18,19,28,29,26,35,39,40] and the references in [10]). From this point of view, the characteristic (or standard ) Sturmian sequence of a particular slope is the one having c k = 0 for all k. This expansion of ω is just one of many possible expansions as an infinite composition of morphisms (see work of Arnoux [37], ArnouxFisher [5], Arnoux-Ferenczi-Hubert [4]).…”

“…We will see below that Ostrowski expansions in the sense of [35] appear in a natural way when one considers a multiplicative version of these expansions.…”

An Exercise on Fibonacci Representations

“…As pointed out by Brown [1], Zeckendorff representations exist in a much larger framework, and goes back at least to Ostrowski [6]. This and related number systems were studied in particular by Fraenkel [4] (see also the references there and in [1]).…”

Lattice point counting and the probabilistic method

“…, a k−1 ] = p k /q k with q 1 = 1, q 2 = a 1 , q n = a n−1 q n−1 + q n−2 for all n ≥ 3. Since q n = a n−1 q n−1 + q n−2 , every positive integer n can be written in the form [Os22]). An analog of the Ostrowski representation of integers can be developed for the representation of the real number β. Write…”