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Higher uniformity of bounded multiplicative functions in short intervals on average. (English) Zbl 1521.11059

This paper contains 12 significant results. Space precludes describing them all. The first result is that for any fixed \(k\in\mathbb{N}\) and \(\theta\in(0,1)\) one has \[ \int_X^{2X}\sup_{\substack{P(t)\in\mathbb{R}[t]\\ \mathrm{deg}(P)\le k}} \left|\sum_{x<n\le x+X^{\theta}}\lambda(n)e(-P(n))\right|dx=o(X^{1+\theta}) \] as \(X\to\infty\), where \(\lambda(n)\) is the Liouville function. In fact one can replace \(\lambda(n)\) by any non-pretentious \(1\)-bounded multiplicative function. Moreover one can replace the polynomial phase \(e(-P(n))\) by a degree \(k\) nilsequence \(\overline{F}(g(n)\Gamma)\). As an application it is shown that if \(s(k)\) is the number of \(k\)-tuples attained by \((\lambda(n+1),\lambda(n+2),\ldots,\lambda(n+k))\) as \(n\) varies, then \(s(k)\gg_A k^A\) for any fixed \(A\). This improves work of R. McNamara [Ergodic Theory Dyn. Syst. 41, No. 10, 3060–3115 (2021; Zbl 1480.37014)] which allowed \(A=2\).
It is also shown that for any fixed \(k\in\mathbb{N}\) and \(\varepsilon>0\) one has \[ \mathbb{E}_{h\le X^{\varepsilon}}\left|\mathbb{E}_{n\le X} \lambda(n+a_1h)\lambda(n+a_2h)\ldots\lambda(n+a_k h)\right|=o(1)\tag{*} \] as \(X\to\infty\), for any distinct \(a_1,\ldots,a_k\in\mathbb{N}\). This can be viewed as progress towards a conjecture of P. Sarnak [Not. South Afr. Math. Soc. 43, No. 2, 89–97 (2012; Zbl 1473.11147)], which would imply that \(s(k)\) grows exponentially.
Indeed there is a more general result than (*), where the shifts are allowed to be polynomials in several variables.

MSC:

11N37 Asymptotic results on arithmetic functions
11B30 Arithmetic combinatorics; higher degree uniformity
37A44 Relations between ergodic theory and number theory

References:

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