THE SPECTRUM OF REAL NUMBERS

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.