The rate of convergence in the central limit theorem for endomorphisms of two-dimensional torus (Q2769695)

Let \(\Omega_2\) be a two-dimensional torus and \(\omega=\|a_{ij} \|\) \((i,j=1,2)\) be the quadratic matrix of second order with integer elements and determinant \(\det\|a_{ij} \|=\pm 1\). Suppose that no eigenvalue of the matrix \(\omega\) is equal to one. The transformation \(Tx=\{x \omega\}= (\{a_{11}x_1 +a_{12}x_2\}\), \(\{a_{21}x_1+ a_{22}x_2\})\), where \(x= (x_1,x_2) \in\Omega_1\), is invertible and preserves the normed Haar measure on \(\Omega_2\) and it is mixing of all degrees. Let \(h\) be a real function defined in the plane \(R\times R\), and suppose that \(h\) is periodic with period one with respect each variable. The author describes the rate of convergence in the central limit theorem for sums \(\sum^{n-1}_{k=0} h(x\omega^k)\).NEWLINENEWLINEFor the entire collection see [Zbl 0968.00043].