A classical theorem of Koksma states that for Lebesgue almost every x > 1 the sequence (x n) n=1 ∞ is uniformly distributed modulo one. In the present paper we extend Koksma’s theorem to the pair correlation setting. More precisely, we show that for Lebesgue almost every x > 1 the pair correlations of the fractional parts of (x n) n=1 ∞ are asymptotically Poissonian. The proof is based on a martingale approximation method.
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