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28d05 (secondary)

Journal of the London Mathematical Society (1)

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ERGODIC AVERAGES FOR INDEPENDENT POLYNOMIALS AND APPLICATIONS
NIKOS FRANTZIKINAKIS, BRYNA KRA
Journal:

Journal of the London Mathematical Society / Volume 74 / Issue 1 / August 2006

Published online by Cambridge University Press:

18 August 2006, pp. 131-142

Print publication:

August 2006
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Szemerédi's theorem states that a set of integers with positive upper density contains arbitrarily long arithmetic progressions. Bergelson and Leibman generalized this, showing that sets of integers with positive upper density contain arbitrarily long polynomial configurations; Szemerédi's theorem corresponds to the linear case of the polynomial theorem. We focus on the case farthest from the linear case, that of rationally independent polynomials. We derive results in ergodic theory and in combinatorics for rationally independent polynomials, showing that their behavior differs sharply from the general situation.