Multiple recurrence and convergence for sequences related to the prime numbers

@inproceedings{Frantzikinakis2006MultipleRA,
  title={Multiple recurrence and convergence for sequences related to the prime numbers},
  author={Nikos Frantzikinakis and Bernard Host and Bryna Kra},
  year={2006}
}
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… 
View PDF on arXiv
59 Citations
Highly Influential Citations
6
Background Citations
24
Methods Citations
8
Results Citations
3

59 Citations

Citation Type

Citation Type

Random differences in Szemerédi’s theorem and related results
We introduce a new, elementary method for studying random differences in arithmetic progressions and convergence phenomena along random sequences of integers. We apply our method to obtain…
Integer part polynomial multiple recurrence along shifted primes
Following an approach presented by N. Frantzikinakis, B. Host and B. Kra, we show that the parameters in the multidimensional Szemer\'edi theorem for integer part polynomials have non-empty…
The polynomial multidimensional Szemerédi Theorem along shifted primes
AbstractIf $\vec q_1 ,...,\vec q_m $ : ℤ → ℤℓ are polynomials with zero constant terms and E ⊂ ℤℓ has positive upper Banach density, then we show that the set E ∩ (E − $\vec q_1 $ (p − 1)) ∩ … ∩ (E…
Nilsequences and multiple correlations along subsequences
  • A. Le
  • Mathematics
    Ergodic Theory and Dynamical Systems
  • 2020
The results of Bergelson, Host and Kra, and Leibman state that a multiple polynomial correlation sequence can be decomposed into a sum of a nilsequence (a sequence defined by evaluating a continuous…
Works Title An Inverse Theorem for the Uniformity Seminorms Associated with the Action of
Let F a finite field. We show that the universal characteristic factor for the Gowers–Host–Kra uniformity seminorm U(X) for an ergodic action (Tg)g∈Fω of the infinite abelian group F on a probability…
J un 2 00 9 AN INVERSE THEOREM FOR THE UNIFORMITY SEMINORMS ASSOCIATED WITH THE ACTION OF F
Let F a finite field. We show that the universal characteristic factor for the Gowers-Host-Kra uniformity seminorm U(X) for an ergodic action (Tg)g∈Fω of the infinite abelian group F on a probability…
Furstenberg Systems of Bounded Multiplicative Functions and Applications
We prove a structural result for measure preserving systems naturally associated with any finite collection of multiplicative functions that take values on the complex unit disc. We show that these…
E-ergodicity and speedups
We introduce the notion of “E-ergodicity” of a measure-preserving dynamical system (where E is a subset of N). We show that given an E-ergodic system T and aperiodic system S, T can be sped up to…
MULTIDIMENSIONAL CONFIGURATIONS IN THE PRIMES WITH SHIFTED PRIME STEPS
Let P denote the set of primes. For a fixed dimension d, CookMagyar-Titichetrakun, Tao-Ziegler and Fox-Zhao independently proved that any subset of positive relative density of P contains an…
Integer part independent polynomial averages and applications along primes
Exploiting the equidistribution properties of polynomial sequences, following the methods developed by Leibman ("Pointwise Convergence of ergodic averages for polynomial sequences of translations on…
...
1
2
3
4
5
...

References

SHOWING 1-10 OF 25 REFERENCES
ASPECTS OF UNIFORMITY IN RECURRENCE
We analyze and cite applications of various, loosely related notions of uni- formity inherent to the phenomenon of (multiple) recurrence in ergodic theory. An assort- ment of results are obtained,…
An ergodic Szemerédi theorem for commuting transformations
The classical Poincar6 recurrence theorem asserts that under the action of a measure preserving transformation T of a finite measure space (X, ~, p.), every set A of positive measure recurs in the…
Linear equations in primes
Consider a system ψ of nonconstant affine-linear forms ψ 1 , ... , ψ t : ℤ d → ℤ, no two of which are linearly dependent. Let N be a large integer, and let K ⊆ [-N, N] d be convex. A generalisation…
Nonconventional ergodic averages and nilmanifolds
We study the L2-convergence of two types of ergodic averages. The first is the average of a product of functions evaluated at return times along arithmetic progressions, such as the expressions…
The primes contain arbitrarily long arithmetic progressions
We prove that there are arbitrarily long arithmetic progressions of primes. There are three major ingredients. The first is Szemeredi�s theorem, which asserts that any subset of the integers of…
Convergence of multiple ergodic averages along polynomials of several variables
AbstractLetT be an invertible measure preserving transformation of a probability measure spaceX. Generalizing a recent result of Host and Kra, we prove that the averages $$T^{p1(u)} f$$ converge…
AN INVERSE THEOREM FOR THE GOWERS U4-NORM
Abstract We prove the so-called inverse conjecture for the Gowers Us+1-norm in the case s = 3 (the cases s < 3 being established in previous literature). That is, we show that if f : [N] → ℂ is a…
Universal characteristic factors and Furstenberg averages
Let X=(X^0,\mu,T) be an ergodic measure preserving system. For a natural number k we consider the averages (*) 1/N \sum_{n=1}^N \prod_{j=1}^k f_j(T^{n a_j}x) where the functions f_j are bounded, and…
ON DIFFERENCE SETS OF SEQUENCES OF INTEGERS . III
where the maximum is taken for those sets ul< u~-K.. , which form an &‘-set relative to the set 12, 22, . . . , n2, . . . . see [ll].) In the case of the arithmetic progressions of three terms, we…
Convergence of polynomial ergodic averages
We prove theL2 convergence for an ergodic average of a product of functions evaluated along polynomial times in a totally ergodic system. For each set of polynomials, we show that there is a…
...
1
2
3
...