Idempotent Ultrafilters and Polynomial Recurrence
نویسندگان
چکیده
In the thirty or so years since H. Furstenberg reproved Szemerédi’s theorem using methods from ergodic theory, many striking discoveries have been made in the area now known as Ergodic Ramsey theory. Perhaps the most surprising of these is the discovery that recurrence results can be obtained for polynomial sets, meaning sets of values of polynomials. The following pretty theorem, a special case of a more general theorem proved by V. Bergelson, H. Furstenberg, and R. McCutcheon in [1], is a typical result in this direction.
منابع مشابه
Idempotent Ultrafilters, Multiple Weak Mixing and Szemerédi’s Theorem for Generalized Polynomials
It is possible to formulate the polynomial Szemerédi theorem as follows: Let qi(x) ∈ Q[x] with qi(Z) ⊂ Z, 1 ≤ i ≤ k. If E ⊂ N has positive upper density then there are a, n ∈ N such that {a, a+q1(n)−q1(0), a+qk(n)−qk(0)} ⊂ E. Using methods of abstract ergodic theory, topological algebra in βN, and some recently-obtained knowledge concerning the relationship between translations on nilmanifolds ...
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