A1 Refereed original research article in a scientific journal
Higher uniformity of bounded multiplicative functions in short intervals on average
Authors: Matomäki Kaisa, Radziwiłł Maksym, Tao Terence, Teräväinen Joni, Ziegler Tamar
Publisher: ANNALS MATHEMATICS, FINE HALL
Publication year: 2023
Journal: Annals of Mathematics
Journal name in source: ANNALS OF MATHEMATICS
Journal acronym: ANN MATH
Volume: 197
Issue: 2
First page : 739
Last page: 857
Number of pages: 119
ISSN: 0003-486X
eISSN: 1939-8980
DOI: https://doi.org/10.4007/annals.2023.197.2.3(external)
Web address : https://doi.org/10.4007/annals.2023.197.2.3(external)
Self-archived copy’s web address: https://research.utu.fi/converis/portal/detail/Publication/179188311(external)
Let lambda denote the Liouville function. We show that, as X-+ oo,2X Zsup X P(Y)ER[Y] degP 0 < theta < 1 fiixed 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)Gamma). By the inverse theory for the Gowers norms this implies the higher order asymptotic uniformity result ZX 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 lambda over short polynomial progressions (n + P1(m), ... , n + Pk(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+epsilon), thus strengthening also previous work on the Fourier uniformity of the Liouville function. 2X l lambda lUk+1([x,x+H]) dx = o(X)
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