Recently, I invented the following theorem and found a proof, it seems strange since it is very counter-intuitive to me. The proof is long and non-conceptual. Is there a place or a branch of math where I can read about it? Or at least a key word? Any explanation is definitely welcome! Sorry for my terminology and notation - I am not a mathematician. The corrections to my notation and terminology are definitely welcome.
Suppose we are given a set $S = \mathbb{N}^{+} \cup \mathbb{N}^{+}x$, where $x$ is a positive irrational number. Prove that there exist $\alpha\in\mathbb{R}$ and $T\in\mathbb{R}^{+}$ such that, for any $k\in\mathbb{N}^+$, the interval $(\alpha + k T, \alpha + k T + T)$ contains exactly one number from $S$.
As Pietro Majer explained in a comment, the result (which is nice) reduces to the statement that for an irrational number $0<T<1$, any $k\in\mathbb{N}^+$ is of the form $\lfloor\frac{n}{T}\rfloor$ or $\lfloor\frac{n}{1-T}\rfloor$ with $n\in\mathbb{N}^+$, but not both. This is known as Rayleigh's theorem or Beatty's theorem, see here.