Matomäki, Kaisa; Radziwiłł, Maksym; Tao, Terence; Teräväinen, Joni; Ziegler, Tamar Higher uniformity of bounded multiplicative functions in short intervals on average. (English) Zbl 1521.11059 Ann. Math. (2) 197, No. 2, 739-857 (2023). 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. Reviewer: D. R. Heath-Brown (Oxford) Cited in 4 Documents MSC: 11N37 Asymptotic results on arithmetic functions 11B30 Arithmetic combinatorics; higher degree uniformity 37A44 Relations between ergodic theory and number theory Keywords:Chowla conjecture; Gowers uniformity; Liouville function; nilsequences; sign patterns Citations:Zbl 1480.37014; Zbl 1473.11147 × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] Helfgott, A. H.; Radziwi{\l}{\l}, M., Expansion, divisibility and parity (2021) [2] Bergelson, V.; Leibman, A., Polynomial extensions of van der {W}aerden’s and {S}zemer\'{e}di’s theorems, J. Amer. Math. Soc.. Journal of the Amer. Math. Soc., 9, 725-753 (1996) · Zbl 0870.11015 · doi:10.1090/S0894-0347-96-00194-4 [3] Blakley, G. R.; Roy, Prabir, A {H}\"{o}lder type inequality for symmetric matrices with nonnegative entries, Proc. Amer. Math. Soc.. Proceedings of the Amer. Math. 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