Abstract
The classical theorem of Riesz and Raikov states that if a > 1 is an integer and ƒ is a function in L 1(ℝ/ℤ), then the averages converge to the mean value of ƒ over [0, 1] for almost every x in [0, 1]. In this paper we prove that, for ƒ in L 1(ℝ/ℤ), the averages A n a ƒ(x) converge a.e. to the integral of ƒ over [0, 1] for almost every a > 1. Furthermore we obtain convergence rates in this strong law of large numbers.
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