A. Leibman

Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold

We show that the orbit of a point on a compact nilmanifold X under the action of a polynomial sequence of translations on X is well distributed on the union of several sub-nilmanifolds of X. This implies that the ergodic averages of a continuous function on X along a polynomial sequence of translations on X converge pointwise.

Set-polynomials and polynomial extension of the Hales-Jewett Theorem

An abstract, Hales-Jewett type extension of the polynomial van der Waerden Theorem [BL] is established: Theorem. Let r, d, q ∈ N. There exists N ∈ N such that for any r-coloring of the set of subsets of V = {1, . . . , N}d×{1, . . . , q} there exist a set a ⊂ V and a nonempty set γ ⊆ {1, . . . , N} such that a∩ (γd × {1, . . . , q}) = ∅, and the subsets a, a ∪ (γd × {1}), a ∪ (γd × {2}), . . ., a ∪ (γd × {q}) are all of the same color. This “polynomial” Hales-Jewett theorem contains refinements of many combinatorial facts as special cases. The proof is achieved by introducing and developing the apparatus of set-polynomials (polynomials whose coefficients are finite sets) and applying the methods of topological dynamics.

Pointwise convergence of ergodic averages for polynomial actions of

Generalizing the one-parameter case, we prove that the orbit of a point on a compact nilmanifold X under a polynomial action of

\mathbb{Z}^{d}

\mathbb{Z}^{d}

by translations on a nilmanifold

by translations on X is uniformly distributed on the union of several sub-nilmanifolds of X . As a corollary we obtain the pointwise ergodic theorem for polynomial actions of

Nilsequences, null-sequences, and multiple correlation sequences

\mathbb{Z}^{d}

Orbit of the diagonal in the power of a nilmanifold

by translations on a nilmanifold.

The shifted primes and the multidimensional Szemerédi and polynomial Van der Waerden theorems

A (d-parameter) basic nilsequence is a sequence of the form \psi(n)=f(a^{n}x), n \in Z^{d}, where x is a point of a compact nilmanifold X, a is a translation on X, and f is a continuous function on X; a nilsequence is a uniform limit of basic nilsequences. If X is a compact nilmanifold, Y is a subnilmanifold of X, g(n) is a (d-parameter) polynomial sequence of translations of X, and f is a continuous function on X, we show that the sequence \int_{g(n)Y}f is the sum of a basic nilsequence and a sequence that converges to zero in uniform density (a null-sequence). We also show that an integral of a family of nilsequences is a nilsequence plus a null-sequence. We deduce that for any invertible finite measure preserving system (W,\mu,T), integer polynomials p_{1},...,p_{k} on Z^{d}, and measurable sets A_{1},...,A_{k} in W, the sequence \mu(T^{p_{1}(n)}A_{1}\cap...\cap T^{p_{k}(n)}A_{k}), n\in Z^{d}, is the sum of a nilsequence and a null-sequence.

Multiple polynomial correlation sequences and nilsequences

Let X be a nilmanifold, that is, a compact homogeneous space of a nilpotent Lie group G, and let a 2 G. We study the closure of the orbit of the diagonal of X r under the action (a p1(n)

Rational sub-nilmanifolds of a compact nilmanifold

Abstract In this short note we establish new refinements of multidimensional Szemeredi and polynomial Van der Waerden theorems along the shifted primes.