A sharp estimate for the rate of convergence in mean of Birkhoff sums for some classes of periodic differentiable functions (Q2498306)

For each vector \((\alpha_1,\alpha_2,\ldots, \alpha_d)\in\mathbb{R}^d\) one defines the shift \(T_\alpha:\mathbb{T}^d\to\mathbb{T}^d\) on the torus \(\mathbb{T}^d\), \(T_\alpha(x)=x+\alpha\,\,(\text{mod}\,\,1)\), i.e., each coordinate of \(x+\alpha\) is computed modulo 1. The Birkhoff sum of order \(n\) over the transformation \(T_\alpha\), \(S_n^\alpha(f)\), associated to a Lebesgue integrable function \(f:\mathbb{T}^d\to\mathbb{R}\) is defined by \[ S_n^\alpha(f)(x)=\sum_{s=0}^{n-1}f\circ T_\alpha^s=\sum_{s=0}^{n-1}f(x+s\alpha). \] The Birkhoff means \({{1}\over{n}}S_n^\alpha(f)\) converge uniformly to the spatial mean \(I(f)=\int_{\mathbb{T}^d}f(x)\, dx\) for each continuous function on \(\mathbb{T}^d\) iff \(\alpha\) is irrational (i.e., its coordinates are independent over \(\mathbb{Z}\)). For each function \(f\in L_p(\mathbb{T}^d)\), \(0<p<\infty\), the Birkhoff means converge to \(I(f)\) in \(L_p\). The author studies the rate of convergence in the space \(L_p\) of the Birkhoff means. Namely, one obtains a sharp estimate on this rate of convergence in the case when \(\alpha\) is badly approximable and \(f\) is an absolutely continuous periodic function of zero mean or a function in the space of the Bessel potentials.