Rigorous diffusion properties for the sawtooth map (Q1189236)

Consider the map \(S:\mathbb{T}^ 2\to\mathbb{T}^ 2\), \(S(A,\theta)=(A+K\theta,A+(K+1)\theta)\pmod{2\pi}\), where \(K>0\), and the torus \(\mathbb{T}^ 2\) is identified with \([-\pi,\pi)^ 2\). The Lebesgue measure on \(\mathbb{T}^ 2\) is invariant under \(S\). It is proved that for each \(t\in [0,1]\) \[ \lim_{K\to\infty}\mathbb{V}([\sqrt{3}/\pi\sqrt{K}]\sum^{[Kt]- 1}_{i=0}S^ i_ 2)=t, \] where \(\mathbb{V}\) denotes the variance with respect to the normalized Lebesgue measure on \(\mathbb{T}^ 2\) and \(S^ i_ 2\) is the second component of the \(i\)'th iterate of \(S\), understood as a real number in \([\pi,\pi)\).