It is known that no general statements can be made about the rate of convergence in Birkhoff’s ergodic theorem. However, it is possible to obtain some results in specific settings. In this talk I will consider weighted sums and discuss their speed of convergence to the mean value. I will provide estimates for such speed of convergence in terms of Diophantine properties of the involved parameters. This talk is based on an ongoing investigation with N. Chalmoukis, L. Colzani and B. Gariboldi.