Nikos Frantzikinakis

Multiple recurrence and convergence for sequences related to the prime numbers

For any measure preserving system (X, , μ,T) and A ∈ with μ(A) > 0, we show that there exist infinitely many primes p such that (the same holds with p − 1 replaced by p + 1). Furthermore, we show the existence of the limit in L 2(μ) of the associated ergodic average over the primes. A key ingredient is a recent result of Green and Tao on the von Mangoldt function. A combinatorial consequence is that every subset of the integers with positive upper density contains an arithmetic progression of length three and common difference of the form p − 1 (or p + 1) for some prime p.

Polynomial averages converge to the product of integrals

We answer a question posed by Vitaly Bergelson, showing that in a totally ergodic system, the average of a product of functions evaluated along polynomial times, with polynomials of pairwise differing degrees, converges inL2 to the product of the integrals. Such averages are characterized by nilsystems and so we reduce the problem to one of uniform distribution of polynomial sequences on nilmanifolds.

Convergence of multiple ergodic averages for some commuting transformations

We prove the

Ergodic averages of commuting transformations with distinct degree polynomial iterates

L^{2}

Ergodic Averages for Independent Polynomials and Applications

convergence for the linear multiple ergodic averages of commuting transformations

Multiple correlation sequences and nilsequences

T_{1}, ..., T_{l}

Pointwise convergence for cubic and polynomial multiple ergodic averages of non-commuting transformations

, assuming that each map

Equidistribution of sparse sequences on nilmanifolds

T_i

Uniformity in the polynomial Wiener–Wintner theorem

and each pair

The structure of strongly stationary systems

T_iT_j^{-1}