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Higher uniformity of bounded multiplicative functions in short intervals on average

Matomäki, Kaisa and Radziwiłł, Maksym and Tao, Terence and Teräväinen, Joni and Ziegler, Tamar (2020) Higher uniformity of bounded multiplicative functions in short intervals on average. . (Unpublished)

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Let λ denote the Liouville function. We show that, as X → ∞, ∫^(2X)_X sup_(P(Y)∈ℝ[Y]deg(P)≤k) ∣ ∑_(x≤n≤x+H) λ(n)e(−P(n))∣ dx = o(XH) for all fixed k and X^θ ≤ H ≤ X with 0 < θ < 1 fixed but arbitrarily small. Previously this was only established for k ≤ 1. We obtain this result as a special case of the corresponding statement for (non-pretentious) 1-bounded multiplicative functions that we prove. In fact, we are able to replace the polynomial phases e(−P(n)) by degree k nilsequences F⎯(g(n)Γ). By the inverse theory for the Gowers norms this implies the higher order asymptotic uniformity result ∫^(2X)_X ‖λ‖_(U^(k+1)([x,x+H])) dx = o(X) in the same range of H. We present applications of this result to patterns of various types in the Liouville sequence. Firstly, we show that the number of sign patterns of the Liouville function is superpolynomial, making progress on a conjecture of Sarnak about the Liouville sequence having positive entropy. Secondly, we obtain cancellation in averages of λ over short polynomial progressions (n + P₁(m),…,n + P_k(m)), which in the case of linear polynomials yields a new averaged version of Chowla's conjecture. We are in fact able to prove our results on polynomial phases in the wider range H ≥ exp ((log X)^(5/8 + ε)), thus strengthening also previous work on the Fourier uniformity of the Liouville function.

Item Type:Report or Paper (Discussion Paper)
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URLURL TypeDescription Paper
Teräväinen, Joni0000-0001-6258-8004
Additional Information:This work was initiated at the American Institute of Mathematics workshop on Sarnak’s conjecture in December 2018. KM was supported by Academy of Finland grant no. 285894. MR acknowledges the support of NSF grant DMS-1902063 and a Sloan Fellowship. TT was supported by a Simons Investigator grant, the James and Carol Collins Chair, the Mathematical Analysis & Application Research Fund Endowment, and by NSF grant DMS-1764034. JT was supported by a Titchmarsh Fellowship. TZ was supported by ERC grant ErgComNum 682150. We thank Amita Malik, Redmond McNamara and Peter Sarnak for helpful discussions.
Funding AgencyGrant Number
Academy of Finland285894
Alfred P. Sloan FoundationUNSPECIFIED
Simons FoundationUNSPECIFIED
James and Carol Collins ChairUNSPECIFIED
Mathematical Analysis and Application Research Fund EndowmentUNSPECIFIED
University of OxfordUNSPECIFIED
European Research Council (ERC)682150
Record Number:CaltechAUTHORS:20210825-184540549
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:110532
Deposited By: George Porter
Deposited On:26 Aug 2021 14:02
Last Modified:26 Aug 2021 14:02

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