A1 Vertaisarvioitu alkuperäisartikkeli tieteellisessä lehdessä
Higher uniformity of bounded multiplicative functions in short intervals on average
Tekijät: Matomäki Kaisa, Radziwiłł Maksym, Tao Terence, Teräväinen Joni, Ziegler Tamar
Kustantaja: ANNALS MATHEMATICS, FINE HALL
Julkaisuvuosi: 2023
Journal: Annals of Mathematics
Tietokannassa oleva lehden nimi: ANNALS OF MATHEMATICS
Lehden akronyymi: ANN MATH
Vuosikerta: 197
Numero: 2
Aloitussivu: 739
Lopetussivu: 857
Sivujen määrä: 119
ISSN: 0003-486X
eISSN: 1939-8980
DOI: https://doi.org/10.4007/annals.2023.197.2.3
Verkko-osoite: https://doi.org/10.4007/annals.2023.197.2.3
Rinnakkaistallenteen osoite: https://research.utu.fi/converis/portal/detail/Publication/179188311
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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