Define $T: \mathbb{R}^2/\mathbb{Z}^2 \to \mathbb{R}^2 / \mathbb{Z}^2$ by $$T(x,y)=(x+\alpha, x+y).$$ Suppose that $\alpha \notin \mathbb{Q}$. By using Fourier series, show that $T$ is ergodic with respect to Lebesgue measure.
Below is the solution to this problem. But I don't understand why the Fourier series of $f\circ T^k$ has the coefficients $a_{(n,m)} e^{2\pi i(nk\alpha + mk(k-1)\alpha /2)}$. I would greatly appreciate if anyone can explain this to me.
