Multiple recurrence and convergence along the primes

Wooley, Trevor D.; Ziegler, Tamar

doi:10.1353/ajm.2012.0048

Cited by 37 publications

(47 citation statements)

References 23 publications

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“…Note that convergence of the averages on the right hand side follows from the next result that was proved in [24] conditional to some conjectures obtained later in [34,35] and the convergence part was also proved in [71]: Theorem 4.3. Let (X, µ, T ) be a system.…”

Section: 12mentioning

confidence: 63%

The logarithmic Sarnak conjecture for ergodic weights

2018

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The Möbius disjointness conjecture of Sarnak states that the Möbius function does not correlate with any bounded sequence of complex numbers arising from a topological dynamical system with zero topological entropy. We verify the logarithmically averaged variant of this conjecture for a large class of systems, which includes all uniquely ergodic systems with zero entropy. One consequence of our results is that the Liouville function has super-linear block growth. Our proof uses a disjointness argument and the key ingredient is a structural result for measure preserving systems naturally associated with the Möbius and the Liouville function. We prove that such systems have no irrational spectrum and their building blocks are infinite-step nilsystems and Bernoulli systems. To establish this structural result we make a connection with a problem of purely ergodic nature via some identities recently obtained by Tao. In addition to an ergodic structural result of Host and Kra, our analysis is guided by the notion of strong stationarity which was introduced by Furstenberg and Katznelson in the early 90's and naturally plays a central role in the structural analysis of measure preserving systems associated with multiplicative functions.

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Section: 12mentioning

confidence: 63%

The logarithmic Sarnak conjecture for ergodic weights

2018

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“…Iterating this, we conclude in particular that H(X k H |Y H ) kH(X H |Y H ) + o A→∞ (1) for any natural numbers k, H with H − H k H H + (note that the number of iterations here is at most H + , so that the o A→∞ (1) error stays under control). From this and (29), (33) we see that…”

mentioning

confidence: 53%

The Logarithmically Averaged Chowla and Elliott Conjectures for Two-Point Correlations

Tao

2016

Forum of Mathematics, Pi

209

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Let λ denote the Liouville function. The Chowla conjecture, in the two-point correlation case, asserts thatas x → ∞, for any fixed natural numbers a 1 , a 2 and nonnegative integer b 1 , b 2 with a 1 b 2 −a 2 b 1 = 0. In this paper we establish the logarithmically averaged versionof the Chowla conjecture as x → ∞, where 1 ω(x) x is an arbitrary function of x that goes to infinity as x → ∞, thus breaking the 'parity barrier' for this problem. Our main tools are the multiplicativity of the Liouville function at small primes, a recent result of Matomäki, Radziwiłł, and the author on the averages of modulated multiplicative functions in short intervals, concentration of measure inequalities, the Hardy-Littlewood circle method combined with a restriction theorem for the primes, and a novel 'entropy decrement argument'. Most of these ingredients are also available (in principle, at least) for the higher order correlations, with the main missing ingredient being the need to control short sums of multiplicative functions modulated by local nilsequences. Our arguments also extend to more general bounded multiplicative functions than the Liouville function λ, leading to a logarithmically averaged version of the Elliott conjecture in the two-point case. In a subsequent paper we will use this version of the Elliott conjecture to affirmatively settle the Erdős discrepancy problem.

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“…The presence of the dilation factor p in the shifts T ph α i in (2.2) is a key feature of this principle that is not present in the classical Furstenberg correspondence principle, and is introduced via the entropy decrement argument from [37]. We remark that the existence of the limit in (2.2) can also be derived from the general convergence results for multiple ergodic averages along the primes in [14], [45], and the logarithmically averaged limit E log p≤P can then be replaced by the ordinary average E p≤P . In fact we have a useful formula for the limit; see Proposition 2.6 below.…”

Section: Notationmentioning

confidence: 95%

Value Patterns of Multiplicative Functions and Related Sequences

Tao

Teräväinen

2019

Forum of Mathematics, Sigma

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We study the existence of various sign and value patterns in sequences defined by multiplicative functions or related objects. For any set A whose indicator function is "approximately multiplicative" and uniformly distributed on short intervals in a suitable sense, we show that the density of the pattern n + 1 ∈ A, n + 2 ∈ A, n + 3 ∈ A is positive, as long as A has density greater than 1 3 . Using an inverse theorem for sumsets and some tools from ergodic theory, we also provide a theorem that deals with the critical case of A having density exactly 1 3 , below which one would need nontrivial information on the local distribution of A in Bohr sets to proceed. We apply our results firstly to answer in a stronger form a question of Erdős and Pomerance on the relative orderings of the largest prime factors P + (n), P + (n + 1), P + (n + 2) of three consecutive integers. Secondly, we show that the tuple (ω(n + 1), ω(n + 2), ω(n + 3)) (mod 3) takes all the 27 possible patterns in (Z/3Z) 3 with positive lower density, with ω(n) being the number of distinct prime divisors. We also prove a theorem concerning longer patterns n + i ∈ Ai, i = 1, . . . k in approximately multiplicative sets Ai having large enough densities, generalising some results of Hildebrand on his "stable sets conjecture". Lastly, we consider the sign patterns of the Liouville function λ and show that there are at least 24 patterns of length 5 that occur with positive upper density. In all of the proofs we make extensive use of recent ideas concerning correlations of multiplicative functions.1 One could also include the n = 0 case here if one wished, although it will not affect our main results.2 For the precise definitions of the various densities used in this paper, as well as the standard arithmetic functions and asymptotic notation, see Subsection 1.7. 1 arXiv:1904.05096v2 [math.NT] 3 Sep 2019 1 d 2 , one can easily show that the Chowla conjectures for the Liouville and Möbius functions are equivalent if one generalises the distinct linear forms n + h1, .. . , n + h k to non-parallel affine forms a1n + h1, . . . , a k n + h k ; we omit the details. 5 We say that f is completely multiplicative if f (mn) = f (m)f (n) for all m, n ∈ N and f (1) = 1. 6 In [42], the proof was written only for f = λ, but the exact same argument works for any completely multiplicative bounded f that is not weakly pretentious.

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Multiple recurrence and convergence along the primes

Cited by 37 publications

References 23 publications

The logarithmic Sarnak conjecture for ergodic weights

The logarithmic Sarnak conjecture for ergodic weights

The Logarithmically Averaged Chowla and Elliott Conjectures for Two-Point Correlations

Value Patterns of Multiplicative Functions and Related Sequences

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