GAZETA MATEMATIC˘A - SSMR

182 ArticoleAsupra teoremei lui Beatty, şirurilor lui Wythoff şicuvântului lui FibonacciAdrian Reisner 1)Abstract. Beatty’s theorem establishes an equivalence relation so thatthe sets {[nα],n ∈ N ∗ } and {[nβ],n ∈ N ∗ } form a disjoint union of N ∗ ,for α, β ∈ R − Q. ThecaseofWythoff ’s pairs is a specific case of Beaty’stheorem with α = ϕ = 1+√ 5(the golden ratio) and allows us to define2the golden semigroup (or Wythoff ’s semigroup) and on the other hand theFibonacci word. The latter, the sturmian characteristic word of slope 1 ρ , 2is also obtained as the fixed point of the morphism σ(0) = 01, σ(1) = 0 inthe free monoid of base A = {0, 1}. Finally, the Fibonacci word allows usto determine a winning strategy for Rufus Issac’s Nim game (or Wythoff ’sgame).Keywords: Beatty theorem, Zeckendorf representation, Ecuaţii Sturm -Liouville, Combinatorics on words.MSC : 11A67, 68R15.I. Partiţiile mulţimii N ∗Fiind dat numărul real α notăm cu E(α) mulţimea:E(α) ={[nα],n ∈N ∗ },unde [α] este partea întreagă a numărului real α. Avem atunci teoremaurmătoare:Teorema 1. Fiind date trei numere reale α 1