scholarly journals Multiple recurrence and convergence for certain averages along shifted primes

2014 ◽  
Vol 35 (5) ◽  
pp. 1592-1609 ◽  
Author(s):  
WENBO SUN
Keyword(s):  
Ergodic Theory ◽  
Correspondence Principle ◽  
Convergence Result ◽  
Multiple Recurrence ◽  
Banach Density ◽  
Shifted Primes

We show that any subset $A\subset \mathbb{N}$ with positive upper Banach density contains the pattern $\{m,m+[n{\it\alpha}],\dots ,m+k[n{\it\alpha}]\}$, for some $m\in \mathbb{N}$ and $n=p-1$ for some prime $p$, where ${\it\alpha}\in \mathbb{R}\setminus \mathbb{Q}$. Making use of the Furstenberg correspondence principle, we do this by proving an associated recurrence result in ergodic theory along the shifted primes. We also prove the convergence result for the associated averages along primes and indicate other applications of these methods.

scholarly journals Closest integer polynomial multiple recurrence along shifted primes

2016 ◽  
Vol 38 (2) ◽  
pp. 666-685 ◽  
Author(s):  
ANDREAS KOUTSOGIANNIS
Keyword(s):  
Ergodic Theory ◽  
Correspondence Principle ◽  
Integer Part ◽  
Multiple Recurrence ◽  
Transference Principle ◽  
Empty Intersection ◽  
Integer Polynomials ◽  
Convergence Results ◽  
Shifted Primes ◽  
Integer Polynomial

Following an approach presented by Frantzikinakis et al [The polynomial multidimensional Szemerédi theorem along shifted primes. Israel J. Math.194(1) (2013), 331–348], we show that the parameters in the multidimensional Szemerédi theorem for closest integer polynomials have non-empty intersection with the set of shifted primes $\mathbb{P}-1$ (or, similarly, of $\mathbb{P}+1$). Using the Furstenberg correspondence principle, we show this result by recasting it as a polynomial multiple recurrence result in measure ergodic theory. Furthermore, we obtain integer part polynomial convergence results by the same method, which is a transference principle that enables one to deduce results for $\mathbb{Z}$-actions from results for flows. We also give some applications of our approach to Gowers uniform sets.

scholarly journals On the distribution of orbits in affine varieties

2014 ◽  
Vol 35 (7) ◽  
pp. 2231-2241
Author(s):  
CLAYTON PETSCHE
Keyword(s):  
Ergodic Theory ◽  
Closed Subset ◽  
Arithmetic Progressions ◽  
Affine Variety ◽  
Arbitrary Characteristic ◽  
Berkovich Spaces ◽  
Density Zero ◽  
Banach Density

Given an affine variety $X$, a morphism ${\it\phi}:X\rightarrow X$, a point ${\it\alpha}\in X$, and a Zariski-closed subset $V$ of $X$, we show that the forward ${\it\phi}$-orbit of ${\it\alpha}$ meets $V$ in at most finitely many infinite arithmetic progressions, and the remaining points lie in a set of Banach density zero. This may be viewed as a weak asymptotic version of the dynamical Mordell–Lang conjecture for affine varieties. The results hold in arbitrary characteristic, and the proof uses methods of ergodic theory applied to compact Berkovich spaces.

scholarly journals Plünnecke inequalities for measure graphs with applications

2015 ◽  
Vol 37 (2) ◽  
pp. 418-439
Author(s):  
KAMIL BULINSKI ◽  
ALEXANDER FISH
Keyword(s):  
Recent Work ◽  
Measure Space ◽  
Correspondence Principle ◽  
Abelian Groups ◽  
Density Estimates ◽  
Vertex Set ◽  
Banach Density ◽  
Measure Preserving Actions

We generalize Petridis’s new proof of Plünnecke’s graph inequality to graphs whose vertex set is a measure space. Consequently, by a recent work of Björklund and Fish, this gives new Plünnecke inequalities for measure-preserving actions which enable us to deduce, via a Furstenberg correspondence principle, Banach density estimates in countable abelian groups that extend those given by Jin.

scholarly journals Plünnecke inequalities for countable abelian groups

2017 ◽  
Vol 2017 (730) ◽  
Author(s):  
Michael Björklund ◽  
Alexander Fish
Keyword(s):  
Abelian Group ◽  
Probability Measure ◽  
Lower Bounds ◽  
Correspondence Principle ◽  
Additive Basis ◽  
Banach Density ◽  
New Form ◽  
Countable Abelian Group ◽  
Product Sets ◽  
Measure Preserving Actions

AbstractWe establish in this paper a new form of Plünnecke-type inequalities for ergodic probability measure-preserving actions of any countable abelian group. Using a correspondence principle for product sets, this allows us to deduce lower bounds on the upper and lower Banach densities of any product set in terms of the upper Banach density of an iterated product set of one of its addends. These bounds are new already in the case of the integers.We also introduce the notion of an ergodic basis, which is parallel, but significantly weaker than the analogous notion of an additive basis, and deduce Plünnecke bounds on their impact functions with respect to both the upper and lower Banach densities on any countable abelian group.

scholarly journals A Quantitative Ergodic Theory Proof of Szemerédi's Theorem

10.37236/1125 ◽  
2006 ◽  
Vol 13 (1) ◽  
Author(s):  
Terence Tao
Keyword(s):  
Ergodic Theory ◽  
Correspondence Principle ◽  
Inverse Theory ◽  
Combinatorial Proof ◽  
Regularity Lemma ◽  
Additive Combinatorics ◽  
Famous Theorem ◽  
Inverse Theorems ◽  
The Fourier Transform ◽  
The Axiom Of Choice

A famous theorem of Szemerédi asserts that given any density $0 < \delta \leq 1$ and any integer $k \geq 3$, any set of integers with density $\delta$ will contain infinitely many proper arithmetic progressions of length $k$. For general $k$ there are essentially four known proofs of this fact; Szemerédi's original combinatorial proof using the Szemerédi regularity lemma and van der Waerden's theorem, Furstenberg's proof using ergodic theory, Gowers' proof using Fourier analysis and the inverse theory of additive combinatorics, and the more recent proofs of Gowers and Rödl-Skokan using a hypergraph regularity lemma. Of these four, the ergodic theory proof is arguably the shortest, but also the least elementary, requiring passage (via the Furstenberg correspondence principle) to an infinitary measure preserving system, and then decomposing a general ergodic system relative to a tower of compact extensions. Here we present a quantitative, self-contained version of this ergodic theory proof, and which is "elementary" in the sense that it does not require the axiom of choice, the use of infinite sets or measures, or the use of the Fourier transform or inverse theorems from additive combinatorics. It also gives explicit (but extremely poor) quantitative bounds.

Ergodic Theory

1983 ◽  
Author(s):  
Karl E. Petersen
Keyword(s):  
Ergodic Theory

Sound as a transverse wave

2017 ◽  
Vol 13 (1) ◽  
pp. 4522-4534
Author(s):  
Armando Tomás Canero
Keyword(s):  
Quantum Mechanics ◽  
Gravitational Waves ◽  
Longitudinal Wave ◽  
Transverse Wave ◽  
Sound Propagation ◽  
Correspondence Principle ◽  
Classical Mechanics ◽  
Wave Model ◽  
Scientific Experiments

This paper presents sound propagation based on a transverse wave model which does not collide with the interpretation of physical events based on the longitudinal wave model, but responds to the correspondence principle and allows interpreting a significant number of scientific experiments that do not follow the longitudinal wave model. Among the problems that are solved are: the interpretation of the location of nodes and antinodes in a Kundt tube of classical mechanics, the traslation of phonons in the vacuum interparticle of quantum mechanics and gravitational waves in relativistic mechanics.

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