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 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.

POLYNOMIAL ROOT SEPARATION

2010 ◽  
Vol 06 (03) ◽  
pp. 587-602 ◽  
Author(s):  
YANN BUGEAUD ◽  
MAURICE MIGNOTTE
Keyword(s):  
Integer Polynomials ◽  
Integer Polynomial

We discuss the following question: How close to each other can two distinct roots of an integer polynomial be? We summarize what is presently known on this and related problems, and establish several new results on root separation of monic, integer polynomials.

ROOTS OF POLYNOMIALS WITH DOMINANT TERM

2011 ◽  
Vol 07 (05) ◽  
pp. 1217-1228 ◽  
Author(s):  
ARTŪRAS DUBICKAS
Keyword(s):  
Constant Coefficient ◽  
Roots Of Unity ◽  
Algebraic Numbers ◽  
Algebraic Integers ◽  
Dominant Term ◽  
Minimal Weight ◽  
Number Systems ◽  
Integer Polynomials ◽  
Roots Of Polynomials ◽  
Integer Polynomial

We characterize all algebraic numbers which are roots of integer polynomials with a coefficient whose modulus is greater than or equal to the sum of moduli of all the remaining coefficients. It turns out that these numbers are zero, roots of unity and those algebraic numbers β whose conjugates over ℚ (including β itself) do not lie on the circle |z| = 1. We also describe all algebraic integers with norm B which are roots of an integer polynomial with constant coefficient B and the sum of moduli of all other coefficients at most |B|. In contrast to the above, the set of such algebraic integers is "quite small". These results are motivated by a recent paper of Frougny and Steiner on the so-called minimal weight β-expansions and are also related to some work on canonical number systems and tilings.

scholarly journals Integer part polynomial correlation sequences

2016 ◽  
Vol 38 (4) ◽  
pp. 1525-1542 ◽  
Author(s):  
ANDREAS KOUTSOGIANNIS
Keyword(s):  
Error Term ◽  
Uniform Density ◽  
Integer Part ◽  
Ergodic Averages ◽  
Multiple Correlation ◽  
Transference Principle ◽  
The Mean ◽  
Correlation Sequences ◽  
Measure Preserving Actions ◽  
Invent Math

Following an approach presented by Frantzikinakis [Multiple correlation sequences and nilsequences. Invent. Math. 202(2) (2015), 875–892], we prove that any multiple correlation sequence defined by invertible measure preserving actions of commuting transformations with integer part polynomial iterates is the sum of a nilsequence and an error term, which is small in uniform density. As an intermediate result, we show that multiple ergodic averages with iterates given by the integer part of real-valued polynomials converge in the mean. Also, we show that under certain assumptions the limit is zero. A transference principle, communicated to us by M. Wierdl, plays an important role in our arguments by allowing us to deduce results for $\mathbb{Z}$-actions from results for flows.

scholarly journals Weighted multiple ergodic averages and correlation sequences

2016 ◽  
Vol 38 (1) ◽  
pp. 81-142 ◽  
Author(s):  
NIKOS FRANTZIKINAKIS ◽  
BERNARD HOST
Keyword(s):  
Bounded Sequence ◽  
Regularity Conditions ◽  
Uniform Density ◽  
Ergodic Averages ◽  
Multiple Correlation ◽  
Mean Convergence ◽  
Integer Polynomials ◽  
Several Variables ◽  
Convergence Results ◽  
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 average of this sequence times any nilsequence converges. Two decomposition results of independent interest play key roles in the proof. 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 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 these 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.

Multiple recurrence and convergence results associated to $${\text{F}}_P^\omega $$ -actions

2015 ◽  
Vol 127 (1) ◽  
pp. 329-378 ◽  
Author(s):  
Vitaly Bergelson ◽  
Terence Tao ◽  
Tamar Ziegler
Keyword(s):  
Multiple Recurrence ◽  
Convergence Results

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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