73 research outputs found
Uniformity in the polynomial Wiener-Wintner theorem
In 1993, E. Lesigne proved a polynomial extension of the Wiener-Wintner
ergodic theorem and asked two questions: does this result have a uniform
counterpart and can an assumption of total ergodicity be replaced by
ergodicity? The purpose of this article is to answer these questions, the first
one positively and the second one negatively.Comment: 12 page
Equidistribution of sparse sequences on nilmanifolds
We study equidistribution properties of nil-orbits when the
parameter is restricted to the range of some sparse sequence that is not
necessarily polynomial. For example, we show that if is a
nilmanifold, is an ergodic nilrotation, and is
positive, then the sequence is equidistributed in
for every . This is also the case when is replaced with ,
where is a function that belongs to some Hardy field, has polynomial
growth, and stays logarithmically away from polynomials, and when it is
replaced with a random sequence of integers with sub-exponential growth.
Similar results have been established by Boshernitzan when is the circle.Comment: 32 pages. References updated, a few small changes made. Appeared in
Journal d'Analyse Mathematiqu
Some open problems on multiple ergodic averages
We survey some recent developments and give a list of open problems regarding
multiple recurrence and convergence phenomena of actions in
ergodic theory and related applications in combinatorics and number theory.Comment: Remarks by the referees incorporated. To appear in the Bulletin of
the Hellenic Mathematical Society. Updates on the status of the problems will
be posted here: http://www.math.uoc.gr/~nikosf/OpenProblems.htm
Multiple recurrence and convergence for Hardy sequences of polynomial growth
We study the limiting behavior of multiple ergodic averages involving
sequences of integers that satisfy some regularity conditions and have
polynomial growth. We show that for "typical" choices of Hardy field functions
with polynomial growth, the averages converge in the mean
and we determine their limit. For example, this is the case if or . Furthermore, if is a
"typical" family of logarithmico-exponential functions of polynomial growth,
then for every ergodic system, the averages converge in the mean
to the product of the integrals of the corresponding functions. For example,
this is the case if the functions are given by different positive
fractional powers of . We deduce several results in combinatorics. We show
that if is a non-polynomial Hardy field function with polynomial growth,
then every set of integers with positive upper density contains arithmetic
progressions of the form . Under suitable
assumptions we get a related result concerning patterns of the form Comment: 42 pages. A correction made in Lemma~5.1. We thank Joanna
Kulaga-Przymus for pointing out this. The remaining text remains unaffecte
The Structure of Strongly Stationary Systems
Motivated by a problem in ergodic Ramsey theory, Furstenberg and Katznelson
introduced the notion of strong stationarity, showing that certain recurrence
properties hold for arbitrary measure preserving systems if they are valid for
strongly stationary ones. We construct some new examples and prove a structure
theorem for strongly stationary systems. The building blocks are Bernoulli
systems and rotations on nilmanifolds
Good weights for the Erd\H{o}s discrepancy problem
The Erd\H{o}s discrepancy problem, now a theorem by T. Tao, asks whether
every sequence with values plus or minus one has unbounded discrepancy along
all homogeneous arithmetic progressions. We establish weighted variants of this
problem, for weights given either by structured sequences that enjoy some
irrationality features, or certain random sequences. As an intermediate result,
we establish unboundedness of weighted sums of bounded multiplicative functions
and products of shifts of such functions. A key ingredient in our analysis for
the structured weights, is a structural result for measure preserving systems
naturally associated with bounded multiplicative functions that was recently
obtained in joint work with B. Host.Comment: 23 pages, a small correction in the statements of Corollary 1.5 and
Proposition 4.
An averaged Chowla and Elliott conjecture along independent polynomials
We generalize a result of Matom\"aki, Radziwi{\l}{\l}, and Tao, by proving an
averaged version of a conjecture of Chowla and a conjecture of Elliott
regarding correlations of the Liouville function, or more general bounded
multiplicative functions, with shifts given by independent polynomials in
several variables. A new feature is that we recast the problem in ergodic terms
and use a multiple ergodic theorem to prove it; its hypothesis is verified
using recent results by Matom\"aki and Radziwi{\l}{\l} on mean values of
multiplicative functions on typical short intervals. We deduce several
consequences about patterns that can be found on the range of various
arithmetic sequences along shifts of independent polynomials.Comment: 15 pages, small changes, to appear in International Mathematics
Research Notice
Convergence of multiple ergodic averages for some commuting transformations
We prove the convergence for the linear multiple ergodic averages of
commuting transformations , assuming that each map and
each pair is ergodic for . The limiting behavior of such
averages is controlled by a particular factor, which is an inverse limit of
nilsystems. As a corollary we show that the limiting behavior of linear
multiple ergodic averages is the same for commuting transformations.Comment: 12 page
A Hardy field extension of Szemeredi's Theorem
In 1975 Szemer\'edi proved that a set of integers of positive upper density
contains arbitrarily long arithmetic progressions. Bergelson and Leibman showed
in 1996 that the common difference of the arithmetic progression can be a
square, a cube, or more generally of the form where is any
integer polynomial with zero constant term. We produce a variety of new results
of this type related to sequences that are not polynomial. We show that the
common difference of the progression in Szemer\'edi's theorem can be of the
form where is any positive real number and denotes
the integer part of . More generally, the common difference can be of the
form where is any function that is a member of a Hardy field
and satisfies and for some
non-negative integer . The proof combines a new structural result for Hardy
sequences, techniques from ergodic theory, and some recent equidistribution
results of sequences on nilmanifolds.Comment: 37 pages. A correction made on the statement of Theorems~B and B'. We
thank Pavel Zorin-Kranich for pointing out that Lemma~4.6 in the previous
version was incorrec
Weighted multiple ergodic averages and correlation sequences
We study mean convergence results for weighted multiple ergodic averages
defined by commuting transformations with iterates given by integer polynomials
in several variables. Roughly speaking, we prove that a bounded sequence is a
good universal weight for mean convergence of such averages if and only if the
averages of this sequence times any nilsequence converge. Key role in the proof
play two decomposition results of independent interest. The first states that
every bounded sequence in several variables satisfying some regularity
conditions is a sum of a nilsequence and a sequence that has small uniformity
norm (this generalizes a result of the second author and B. Kra); and the
second states that every multiple correlation sequence in several variables is
a sum of a nilsequence and a sequence that is small in uniform density (this
generalizes a result of the first author). Furthermore, we use the previous
results in order to establish mean convergence and recurrence results for a
variety of sequences of dynamical and arithmetic origin and give some
combinatorial implications.Comment: 53 pages, small changes made in light of comments from the referee,
to appear in Ergodic Theory and Dynamical System
- β¦