THE SPECTRUM OF REAL NUMBERS - Bobojonov Hamid Hayitboyevich; Khamid Bobojonov; Hayitboyeva Hayriniso Hamid qizi; Khayitboyeva Hayriniso; Hayitboyeva Mo’tabarxon Hamid qizi; Khayitboyeva Mo’tabarxon (Bright Mind Publishing, 2026)
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.
Keywords: Beatty sequence, spectrum, irrational numbers, complementary sequences, golden ratio, floor function, number theory.