Polynomial Extensions of Van Der Waerden’s and Szemerédi’s Theorems
نویسندگان
چکیده
An extension of the classical van der Waerden and Szemerédi the-orems is proved for commuting operators whose exponents are polynomials.As a consequence, for example, one obtains the following result: Let S ⊆ Zlbe a set of positive upper Banach density, let p1(n), . . . , pk(n) be polynomialswith rational coefficients taking integer values on the integers and satisfyingpi(0) = 0, i = 1, . . . , k; then for any v1, . . . , vk ∈ Zl there exist an integer nand a vector u ∈ Zl such that u+ pi(n)vi ∈ S for each i ≤ k.Department of Mathematics, Ohio State University, Columbus, Ohio 43210E-mail address: [email protected] Department of Mathematics, Technion, Haifa 23000, IsraelE-mail address: [email protected] address: Department of Mathematics, Stanford University, Stanford, California 94305E-mail address: [email protected] License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use
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