Man walking along a circle falling in a ditch.

Question

Consider a circle as in the figure. It has a small ditch of width $L$. A man is walking around the circle with step length $\alpha$ (measured along the circumference). $\alpha$ is irrational. We need to prove that sooner or later he will step into the ditch no matter at which point of the circle he starts at, or how small $L$ is.

I firstly tries this with pegion-hole principle getting nowhere. Then I thought that if he can't step in the length $L$ of the ditch, then he can't also step in an arc of length $L+\alpha$ and so on. But could not come to a conclusion.

Answer 1

What you need to prove is that $\{n\alpha\bmod 1:n\in\Bbb N\}$ is dense in $[0,1]$, where $$x\bmod 1=x-\lfloor x\rfloor$$ is the fractional part of $x$. It can be done quite efficiently using the pigeonhole principle; there’s a concise version of such a proof here, which I’ll offer in lieu of a hint: it’s concise enough that you may have to put in some effort to verify all of the steps.