# Application of Jacobi's Theorem in Box Principle

Today I was going through Problem Solving Strategies by Arthur Engel, and found this in the chapter Box Principle

Before the question it says it "treats a theorem of Jacobi and its applications"

There is a circle with length ${1}$. A man with irrational step length ${\alpha}$ (measured along the circumference) walks around the circle. The circle has a ditch of width ${\varepsilon > 0}$ on its circumference. Prove that sooner or later, he will step into the ditch no matter how small $\varepsilon$ is.

I don't see how I can link Jacobi's theorem to it. When I searched wikipedia for Jacobi's theorem, I was greeted with $4$ "Jacobi's Theorem". So I don't know which one to use.

## 1 Answer

The author is referring to the following theorem:

If $\alpha$ is irrational, the sequence $n\alpha - \lfloor n \alpha \rfloor$ ($n \in \mathbb{N}$) is dense in $[0,1]$.

Let $S_r \subset \mathbb{R}^2$ be the circle of radius $r$ centered at the origin. Since the interval $[0,1)$ can be identified with $S_r$ via the map $x \mapsto (r\cos(2\pi x), r\sin(2\pi x))$, the theorem above is equivalent to

If $\alpha$ is irrational, the sequence $(r\cos(2\pi n \alpha), r\sin(2\pi n \alpha))$ ($n \in \mathbb{N}$) is dense in $S_r$.

The Wikipedia article about irrational rotations states (an equivalent version of) this result, but doesn't mention Jacobi's name.