THE SPECTRUM OF REAL NUMBERS
Keywords:
Beatty sequence, spectrum, irrational numbers, complementary sequences, golden ratio, floor function, number theory.Abstract
This paper studies the properties of the spectrum S(α) = nα : n 1 generated by a real number α. We prove the uniqueness theorem for spectra, give a criterion for determining whether a natural number belongs to a given spectrum, and establish necessary and sufficient conditions for partitioning the natural numbers into two complementary spectra. Special attention is devoted to the spectra associated with the golden ratio φ. The article also includes solved examples and exercises for independent work.
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