Generalization of Steinhaus conjecture

H. Steinhaus proposed a conjecture about irrational numbers, called the Steinhaus conjecture. Sós et al. proved it, and the results of the study are referred to as the “Three Gap Theorem.” In this paper, we promote the Steinhaus conjecture and prove the generalization of the Three Gap Theorem, in which the definition domain of independent variables is extended from natural number sets to integer sets, and an application of the extended conclusion in the field of music art is illustrated.