Randall McCutcheon

IP-sets and polynomial recurrence

We combine recurrence properties of polynomials and IP-sets and show that polynomials evaluated along IP-sequences also give rise to Poincare sets for measure-preserving systems, that is, sets of integers along which the analogue of the Poincare recurrence theorem holds. This is done by applying to measure-preserving transformations a limit theorem for products of appropriate powers of a commuting family of unitary operators.

Elemental Methods in Ergodic Ramsey Theory

Ramsey theory and topological dynamics.- Infinitary Ramsey theory.- Density Ramsey theory.- Three ergodic roth theorems.- Two Szemeredi theorems.

IP-systems, generalized polynomials and recurrence

Let Ω be an abelian group. A set R ⊂ Ω is a set of recurrence if for any probability measure preserving system (X,B, μ, {Tg}g∈Ω) and any A ∈ A with μ(A) > 0, μ(A ∩ TgA) > 0 for some g ∈ R. If (xi)i=1 is a sequence in Ω, the set of its finite sums {xi1 + xi2 + · · · + xik : i1 < i2 < · · · < ik} is called an IP-set. In [BFM] it is shown that if p : Z → Z is a polynomial vanishing at zero and F is an IP-set in Z then {p(n) : n ∈ F} is a set of recurrence in Z. Here we extend this result to an analagous family of generalized polynomials, that is functions formed from regular polynomials by iterated use of the greatest integer function, as a consequence of a theorem establishing a much wider class of recurrence sets occuring in any (possibly non-finitely generated) abelian group. While these sets do in a sense have a distinctively “polynomial” nature, this far-ranging class includes, even in Z, such examples as {i,j∈α,i

Polynomial Szemerédi theorems for countable modules over integral domains and finite fields

AbstractGiven a pair of vector spacesV andW over a countable fieldF and a probability spaceX, one defines apolynomial measure preserving action ofV onX to be a compositionT o ϕ, where ϕ:V→W is a polynomial mapping andT is a measure preserving action ofW onX. We show that the known structure theory of measure preserving group actions extends to polynomial actions and establish a Furstenberg-style multiple recurrence theorem for such actions. Among the combinatorial corollaries of this result are a polynomial Szemerédi theorem for sets of positive density in finite rank modules over integral domains, as well as the following fact:Let

An Infinitary Polynomial van der Waerden Theorem

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Partition Theorems for Left and Right Variable Words

be a finite family of polynomials with integer coefficients and zero constant term. For any α>0, there exists N ∈ ℕ such that whenever F is a field with |F|≥N and E ⊆F with |E|/|F|≥α, there exist u∈F, u≠0, and w∈E such that w+ϕ(u)∈E for all ϕ∈

Generalized Polynomials and Mild Mixing

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