Abstract: In this paper we define an equivalence relation on the set of positive irrational numbers less than 1. The relation is defined by means of continued fractions. Equivalence classes under this relation are determined by the places of some elements equal to 1 (called essential 1's) in the continued fraction expansion of numbers. Analysis of suprema of all equivalence classes leads to a solution which involves Fibonacci numbers and constitutes the main result of this paper. The problem has its origin in the author's research on the construction of digital lines and upper and lower mechanical and characteristic words according to the hierarchy of runs.
Keywords: Irrational number, equivalence relation, continued fraction, Fibonacci numbers, digital line, (upper, lower) mechanical word, characteristic word, Beatty sequences, Sturmian word, hierarchy, run
Classification (MSC2000): 11A55; 11B39, 03E20, 68R15
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