A Sharp Estimate for the Rate of Convergence in Mean of Birkhoff Sums for Some Classes of Periodic Differentiable Functions

For a badly approximable vector $\alpha$, we obtain a sharp estimate for the rate of convergence in the space $L_p$ ($0<p<\infty$) of the Birkhoff means $\frac1{n}\sum_{s=0}^{n-1} f(x+s\alpha)$ for an absolutely continuous periodic function $f$ and for functions in spaces of Bessel potentials.