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Weighted multiple ergodic averages and correlation sequences
No full textInternational audienceWe 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
Joint ergodicity of fractional powers of primes
Full text linkWe establish mean convergence for multiple ergodic averages with iterates
given by distinct fractional powers of primes and related multiple recurrence
results. A consequence of our main result is that every set of integers with
positive upper density contains patterns of the form , where are positive non-integers and denotes the
-th prime, a property that fails if or is a natural number. Our
approach is based on a recent criterion for joint ergodicity of collections of
sequences and the bulk of the proof is devoted to obtaining good seminorm
estimates for the related multiple ergodic averages. The input needed from
number theory are upper bounds for the number of prime -tuples that follow
from elementary sieve theory estimates and equidistribution results of
fractional powers of primes in the circle.Comment: 27 pages. Referee's comments incorporated. To appear in Forum of
Mathematics, Sigm
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Arbeitsgemeinschaft: Ergodic Theory and Combinatorial Number Theory
Full text linkThe aim of this Arbeitsgemeinschaft was to introduce young researchers with various backgrounds to the multifaceted and mutually perpetuating connections between ergodic theory, topological dynamics, combinatorics, and number theory
Joint ergodicity of sequences
Full text linkA collection of integer sequences is jointly ergodic if for every ergodic
measure preserving system the multiple ergodic averages, with iterates given by
this collection of sequences, converge in the mean to the product of the
integrals. We give necessary and sufficient conditions for joint ergodicity
that are flexible enough to recover most of the known examples of jointly
ergodic sequences and also allow us to answer some related open problems. An
interesting feature of our arguments is that they avoid deep tools from ergodic
theory that were previously used to establish similar results. Our approach is
primarily based on an ergodic variant of a technique pioneered by Peluse and
Prendiville in order to give quantitative variants for the finitary version of
the polynomial Szemer\'edi theorem.Comment: 44 pages, referees comments incorporated, to appear in the Advances
in Mathematic
Going Beyond Counting First Authors in Author Co-citation Analysis
Full text linkThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
Asymptotics for multilinear averages of multiplicative functions
No full textInternational audienceA celebrated result of Hal\'asz describes the asymptotic behavior of the arithmetic mean of an arbitrary multiplicative function with values on the unit disc. We extend this result to multilinear averages of multiplicative functions providing similar asymptotics, thus verifying a two dimensional variant of a conjecture of Elliott. As a consequence, we get several convergence results for such multilinear expressions, one of which generalizes a well known convergence result of Wirsing. The key ingredients are a recent structural result for bounded multiplicative functions proved by the authors and the mean value theorem of Hal\'asz
Degree lowering for ergodic averages along arithmetic progressions
Full text linkWe examine the limiting behavior of multiple ergodic averages associated with
arithmetic progressions whose differences are elements of a fixed integer
sequence. For each , we give necessary and sufficient conditions under
which averages of length of the aforementioned form have the same limit
as averages of -term arithmetic progressions. As a corollary, we derive a
sufficient condition for the presence of arithmetic progressions with length
and restricted differences in dense subsets of integers. These results
are a consequence of the following general theorem: in order to verify that a
multiple ergodic average is controlled by the degree Gowers-Host-Kra
seminorm, it suffices to show that it is controlled by some Gowers-Host-Kra
seminorm, and that the degree control follows whenever we have degree
control. The proof relies on an elementary inverse theorem for the
Gowers-Host-Kra seminorms involving dual functions, combined with novel
estimates on averages of seminorms of dual functions. We use these estimates to
obtain a higher order variant of the degree lowering argument previously used
to cover averages that converge to the product of integrals.Comment: 39 pages. Referee's comments incorporated. To appear in Journal
d'Analyse Mathematiqu
Higher order Fourier analysis of multiplicative functions and appliactions
No full textInternational audienceWe prove a structure theorem for multiplicativefunctions whichstates that an arbitrary multiplicative function of modulus at most can be decomposedinto two terms, one that is approximately periodic and another that has small Gowers uniformity norm of an arbitrary degree. The proof uses tools from higher order Fourier analysis and finitary ergodic theory, and some soft number theoretic input that comes in the form of an orthogonality criterion of K\'atai.We use variants of this structure theorem to derive applications ofnumber theoretic and combinatorialflavor: we give simple necessary and sufficient conditionsfor the Gowers norms (over ) of abounded multiplicative function to be zero, generalizing a classical result of Daboussi we prove asymptotic orthogonality ofmultiplicative functions to ``irrational'' nilsequences, we prove that for certain polynomials in two variables all ``aperiodic'' multiplicative functionssatisfy Chowla's zero mean conjecture, we give the first partition regularity results for homogeneous quadratic equations in three variables,showing for example that on every partition of the integers into finitely many cells there exist distinct belonging to the samecell and such that and the same holds for the equation
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