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We study the convergence of N ⁻¹ ∑ f _1( T a_1n x ) f _2( T a_2n x ) f _3( T a_3n x ), for a measure-preserving system ( X , B, μ, T ) and f ₁, f ₂, f ₃ ∈ L ∞(μ). This generalizes the theorem of Conze and Lesigne on such expressions and simplifies the proof. We also obtain a description of the limit.
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Host et al. (2001) studied this question.
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