Recently, I composed the following math problem and found a solution, it seems strange since it is very counter-intuitive to me. 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 - I am not a mathematician.
Suppose we are given a set $S = \{i\} \cup \{j \cdot x\},$ where $i, j \in \mathbb{N}^{+}$ and $x$ is a positive irrational number. Prove that there exist two real numbers $\alpha$ and positive $T$ such that, for any $k\in\mathbb{N}^+$, the interval $(\alpha + k \cdot T, \alpha + k \cdot T + T),$ contains exactly one number from $S$.
The result is closely related to Beatty sequences. For more details see here.
P.S. I just noticed Joseph Malkevitch made a similar comment an hour ago.