Geodesic
A Sharp Estimate for the Rate of Convergence in Mean of Birkhoff Sums for Some Classes of Periodic Differentiable Functions
Funkcionalʹnyj analiz i ego priloženiâ, Tome 40 (2006) no. 1, pp. 43-51.

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For a badly approximable vector $\alpha$, we obtain a sharp estimate for the rate of convergence in the space $L_p$ ($0$) 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.
Keywords: Birkhoff sum, badly approximable vector, generalized Bessel potential. 
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A. V. Rozhdestvenskii. A Sharp Estimate for the Rate of Convergence in Mean of Birkhoff Sums for Some Classes of Periodic Differentiable Functions. Funkcionalʹnyj analiz i ego priloženiâ, Tome 40 (2006) no. 1, pp. 43-51. http://geodesic.mathdoc.fr/item/FAA_2006_40_1_a3/

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