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Polynomial multiple recurrence over rings of integers

Published online by Cambridge University Press:  06 February 2015

VITALY BERGELSON and
DONALD ROBERTSON
VITALY BERGELSON
Affiliation:
Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA email robertson@math.ohio-state.edu
DONALD ROBERTSON
Affiliation:
Department of Mathematics, The Ohio State University, Columbus, OH 43210, USA email robertson@math.ohio-state.edu
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Abstract

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We generalize the polynomial Szemerédi theorem to intersective polynomials over the ring of integers of an algebraic number field, by which we mean polynomials having a common root modulo every ideal. This leads to the existence of new polynomial configurations in positive-density subsets of $\mathbb{Z}^{m}$ and strengthens and extends recent results of Bergelson, Leibman and Lesigne [Intersective polynomials and the polynomial Szemerédi theorem. Adv. Math.219(1) (2008), 369–388] on polynomials over the integers.

Information

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 36 , Issue 5 , August 2016, pp. 1354 - 1378
Copyright
© Cambridge University Press, 2015 

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