Quantitative bounds in the nonlinear Roth theorem

Peluse, Sarah; Prendiville, Sean

doi:10.48550/arxiv.1903.02592

Cited by 13 publications

(46 citation statements)

References 18 publications

(45 reference statements)

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“…Moreover, in Corollary 1.3 we show that our main result has some far-reaching consequences related to equidistribution properties of general sequences on nilmanifolds. Our approach is to adapt and utilize in our ergodic theory setup, a technique developed by Peluse [35] and Peluse and Prendiville [37] (see also [39] for an exposition of the technique) and used in order to establish quantitative results for finitary variants of special cases of the polynomial Szemerédi theorem. Although the main ideas are elementary, they are a bit cumbersome to implement in full generality, so in order to facilitate reading, we first present our argument in the simpler case of the multiple ergodic averages…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…It is a classical result of Furstenberg and Weiss that these averages converge in L 2 (µ), to the product of the integrals for totally ergodic systems, and that the rational Kronecker factor is a characteristic factor for the above averages. We present in Section 2 a proof of this result using the general principles of the argument used in [37], but also take advantage of various simplifications that our infinitary setup allows (the complete argument is only a few pages long). Subsequently, in Sections 3 and 4 we extend this method to give a proof of our main result, Theorem 1.1, which gives necessary and sufficient conditions for joint ergodicity.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

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Joint ergodicity of sequences

Frantzikinakis¹

2021

Preprint

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A 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 Szemerédi theorem.

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Section: Introduction and Main Resultsmentioning

confidence: 99%

Section: Introduction and Main Resultsmentioning

confidence: 99%

Joint ergodicity of sequences

Frantzikinakis¹

2021

Preprint

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“…Our argument depends upon a convenient criterion for joint ergodicity that was established recently in [7] (and was motivated by work in [23,24]). In order to state it we need to review the definition of the ergodic seminorms from [15].…”

Section: Proof Strategymentioning

confidence: 99%

“…It follows that in (24) when computing a i (h, (n + s/p(h)) 1/d ) we can replace n + s/p(h) with n in the non-linear monomials; this will lead to some error sequences that are 1-bounded for large enough N and can be handled by appealing to Lemma 3.6 (and redefining the sequence z N,h (n)). With this in mind, it follows that in (24) we can replace a i (h, (n + s/p(h))…”

Section: Seminorm Estimates -Sublinear Casementioning

confidence: 99%

Joint ergodicity of fractional powers of primes

Frantzikinakis¹

2021

Preprint

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We 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 {m, m+[p a n ], m+[p b n ]}, where a, b are positive non-integers and pn denotes the n-th prime, a property that fails if a or b 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 k-tuples that follow from elementary sieve theory estimates and equidistribution results of fractional powers of primes in the circle.

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“…Inspired by an argument of Peluse [24] and Peluse and Prendiville [25], Frantzikinakis shows in [17] that a family of sequences a 1 , . .…”

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confidence: 99%

Polynomial ergodic averages for certain countable ring actions

Best¹,

Moragues

2022

DCDS

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A recent result of Frantzikinakis in [<xref ref-type="bibr" rid="b17">17</xref>] establishes sufficient conditions for joint ergodicity in the setting of <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{Z} $\end{document}</tex-math></inline-formula>-actions. We generalize this result for actions of second-countable locally compact abelian groups. We obtain two applications of this result. First, we show that, given an ergodic action <inline-formula><tex-math id="M2">\begin{document}$ (T_n)_{n \in F} $\end{document}</tex-math></inline-formula> of a countable field <inline-formula><tex-math id="M3">\begin{document}$ F $\end{document}</tex-math></inline-formula> with characteristic zero on a probability space <inline-formula><tex-math id="M4">\begin{document}$ (X,\mathcal{B},\mu) $\end{document}</tex-math></inline-formula> and a family <inline-formula><tex-math id="M5">\begin{document}$ \{p_1,\dots,p_k\} $\end{document}</tex-math></inline-formula> of independent polynomials, we have<disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \lim\limits_{N \to \infty} \frac{1}{|\Phi_N|}\sum\limits_{n \in \Phi_N} T_{p_1(n)}f_1\cdots T_{p_k(n)}f_k\ = \ \prod\limits_{j = 1}^k \int_X f_i \ d\mu, $\end{document} </tex-math></disp-formula>where <inline-formula><tex-math id="M6">\begin{document}$ f_i \in L^{\infty}(\mu) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ (\Phi_N) $\end{document}</tex-math></inline-formula> is a Følner sequence of <inline-formula><tex-math id="M8">\begin{document}$ (F,+) $\end{document}</tex-math></inline-formula>, and the convergence takes place in <inline-formula><tex-math id="M9">\begin{document}$ L^2(\mu) $\end{document}</tex-math></inline-formula>. This yields corollaries in combinatorics and topological dynamics. Second, we prove that a similar result holds for totally ergodic actions of suitable rings.

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Quantitative bounds in the nonlinear Roth theorem

Cited by 13 publications

References 18 publications

Joint ergodicity of sequences

Joint ergodicity of sequences

Joint ergodicity of fractional powers of primes

Polynomial ergodic averages for certain countable ring actions

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