Reflection of a Light Ray in a Square and the Distribution of $\{\langle n\gamma\rangle\}_{n=1}^\infty$ for $\gamma\in\mathbb R$

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This post concerns the geometric interpretation of the distribution properties of $\{\langle n\gamma\rangle\}_{n=1}^\infty$ for $\gamma\in\mathbb R$, as discussed in Chapter $4$, Applications of Fourier Series in Stein & Shakarchi's Fourier Analysis.

Suppose that the sides of a square are reflecting mirrors and that a ray of light leaves a point inside the square. What kind of path will the light trace out?

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To solve this problem, the main idea is to consider the grid of the plane formed by successively reflecting the initial square across its sides. With an appropriate choice of axis, the path traced by the light in the square corresponds to the straight line $P + (t, \gamma t)$ in the plane. As a result, the reader may observe that the path will be either closed and
periodic, or it will be dense in the square. The first of these situations will happen if and only if the slope $\gamma$ of the initial direction of the light (determined with respect to one of the sides of the square) is rational. In the second situation, when $\gamma$ is irrational, the density follows from Kronecker's theorem. What stronger conclusion does one get from the
equidistribution theorem?

  1. How does the path traced by the light ray correspond to the straight line $P + (t, \gamma t)$ in the plane $\mathbb R^2$?

Once I understand this, I should be able to approach further questions, namely:

  1. When is the path closed and periodic? When is it dense in the square?

  2. What stronger conclusion follows from the equidistribution theorem?

Equidistribution theorem: If $\gamma\notin \mathbb Q$, then the sequence of fractional parts $\{\langle n\gamma\rangle\}_{n=1}^\infty$ is equidistributed in $[0,1)$.

Notation:

  • $\langle x\rangle$ denotes the fractional part of $x\in \mathbb R$, i.e. $x = \lfloor x\rfloor + \langle x\rangle$.

Let me know if any other details are needed. Thank you!

Best Answer

Rather than reflecting the light ray against the wall it first strikes, reflect the square room across that wall, so that the light ray appears to extend as a straight line into the next reflected square room. Continue indefinitely.

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