Sarnak's Conjecture from the Ergodic Theory Point of View

Kułaga-Przymus, Joanna; Lemańczyk, Mariusz

doi:10.48550/arxiv.2009.04757

“…A key feature of this result is that the interval length h can grow arbitrarily slowly as a function of X. This result has had countless applications to a variety of problems across mathematics, including to partial results towards Chowla's conjecture on correlations of the Liouville function [37], [39], the resolution of the famous Erdős discrepancy problem [36], and progress on Sarnak's Möbius disjointness conjecture (e.g., [38], [3]; see [20] for a more exhaustive list). Since [26], the result has been extended and generalized in various directions.…”

Section: Introduction and Main Resultsmentioning

confidence: 94%

Divisor-bounded multiplicative functions in short intervals

Mangerel¹

2021

Preprint

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We extend the Matomäki-Radziwi l l theorem to a large collection of unbounded multiplicative functions that are uniformly bounded, but not necessarily bounded by 1, on the primes. Our result allows us to estimate averages of such a function f in typical intervals of length h(log X) c , with h = h(X) → ∞ and where c = c f ≥ 0 is determined by the distribution of {|f (p)|}p in an explicit way. We give two applications. Our first application shows that the classical Rankin-Selberg-type asymptotic formula for partial sums of |λ f (n)| 2 , where {λ f (n)}n is the sequence of normalized Fourier coefficients of a primitive non-CM holomorphic cusp form, persists in typical short intervals of length h log X, if h = h(X) → ∞.Our second application shows that the (non-multiplicative) Hooley ∆-function has average value ≫ log log X in typical short intervals of length (log X) 1/2+η , where η > 0 is fixed.

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“…This was essential in his proof of the Erdős discrepancy problem [12], and also enabled him to obtain a logarithmic density analogue of the case k = 2 of Chowla's conjecture. It has also been pivotal in the various developments towards Sarnak's conjecture on the disjointness of the Liouville function from zero entropy dynamical systems (see [13] for a survey).…”

Section: Matomäki-radziwiłł Type Theorems For Additive Functionsmentioning

confidence: 99%

Additive functions in short intervals, gaps and a conjecture of Erdős

Mangerel

¹

2022

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With the aim of treating the local behaviour of additive functions, we develop analogues of the Matomäki–Radziwiłł theorem that allow us to approximate the average of a general additive function over a typical short interval in terms of a corresponding long average. As part of this treatment, we use a variant of the Matomäki–Radziwiłł theorem for divisor-bounded multiplicative functions recently proven in Mangerel (Divisor-bounded multiplicative functions in short intervals. arXiv: 2108.11401). We consider two sets of applications of these methods. Our first application shows that for an additive function $${\varvec{g:}} \mathbb {N} \rightarrow \mathbb {C}$$ g : N → C any non-trivial savings in the size of the average gap $$|{\varvec{g}}{} {\textbf {(}}{\varvec{n}}{} {\textbf {)}}-{\varvec{g}}{} {\textbf {(}}{\varvec{n}}-{\textbf {1}}{} {\textbf {)}} |$$ | g ( n ) - g ( n - 1 ) | implies that $${\varvec{g}}$$ g must have a small first centred moment i.e. the discrepancy of $${\varvec{g}}{} {\textbf {(}}{\varvec{n}}{} {\textbf {)}}$$ g ( n ) from its mean is small on average. We also obtain a variant of such a result for the second moment of the gaps. This complements results of Elliott and of Hildebrand. As a second application, we make partial progress on an old question of Erdős relating to characterizing constant multiples of $${{\textbf {log}}} \,{\varvec{n}}$$ log n as the only almost everywhere increasing additive functions. We show that if an additive function is almost everywhere non-decreasing then it is almost everywhere well approximated by a constant times a logarithm. We also show that if the set $$\{{\varvec{n}} \in \mathbb {N} : {\varvec{g}}{} {\textbf {(}}{\varvec{n}}{} {\textbf {)}} < {\varvec{g}}{} {\textbf {(}}{\varvec{n}}-{\textbf {1}}{} {\textbf {)}}\}$$ { n ∈ N : g ( n ) < g ( n - 1 ) } is sufficiently sparse, and if $${\varvec{g}}$$ g is not extremely large too often on the primes (in a precise sense), then $${\varvec{g}}$$ g is identically equal to a constant times a logarithm.

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“…This was essential in his proof of the Erdős discrepancy problem [28], and also enabled him to obtain a logarithmic density analogue of the case k = 2 of Chowla's conjecture. It has also been pivotal in the various developments towards Sarnak's conjecture on the disjointness of the Liouville function from zero entropy dynamical systems (see [20] for a survey). Our first main result establishes an ℓ 1 -averaged comparison theorem for short and long averages of additive functions, inspired by the theorem of Matomäki and Radziwi l l.…”

Section: Introductionmentioning

confidence: 99%

Additive functions in short intervals, gaps and a conjecture of Erdős

Mangerel

2021

Preprint

0

View full text Add to dashboard Cite

With the aim of treating the local behaviour of additive functions, we develop analogues of the Matomäki-Radziwi l l theorem that allow us to approximate the average of a general additive function over a typical short interval in terms of a corresponding long average. As part of this treatment, we use a variant of the Matomäki-Radziwi l l theorem for divisor-bounded multiplicative functions recently proven in [21]. We consider two sets of applications of these methods. Our first application shows that for an additive function g : N → C any non-trivial savings in the size of the average gap |g(n) − g(n − 1)| implies that g must have a small first centred moment, i.e., the discrepancy of g(n) from its mean is small on average. We also obtain a variant of such a result for the second moment of the gaps. This complements results of Elliott and of Hildebrand. As a second application, we make partial progress on an old question of Erdős relating to characterizing constant multiples of log n as the only almost everywhere increasing additive functions. We show that if an additive function is almost everywhere non-decreasing then it is almost everywhere wellapproximated by a constant times a logarithm. We also show that if the set {n ∈ N : g(n) < g(n−1)} is sufficiently sparse, and if g is not extremely large too often on the primes (in a precise sense), then g is identically equal to a constant times a logarithm.

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Sarnak's Conjecture from the Ergodic Theory Point of View

Cited by 6 publications

References 48 publications

Divisor-bounded multiplicative functions in short intervals

Divisor-bounded multiplicative functions in short intervals

Additive functions in short intervals, gaps and a conjecture of Erdős

Additive functions in short intervals, gaps and a conjecture of Erdős

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