On polynomials in primes and J. Bourgain's circle method approach to ergodic theorems II

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Department of Pure Mathematics, University of Liverpool, P.O. Box 147, Liverpool L69, 3BX, U.K

We show that if q is greater than one, T is a measure preserving transformation of the measure space (X,β,μ) and f is in

L^{q} (X, β, μ)

then if ϕ is a non-constant polynomial mapping the natural numbers to themselves, the averages

π_{N}^{- 1} \sum_{1 \leq p \leq N} f (T^{ϕ (p)} x) (N = 1, 2, ...) c o n v e r \geq μ a l m o s t e v e r y w h e r e . H e r e p r u n s o v e r t h e' s and

π_N!$! denotes their number in [1, N].

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