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Published March 2023 | public
Journal Article

Higher uniformity of bounded multiplicative functions in short intervals on average


Let λ denote the Liouville function. We show that, as X → ∞, ∫^(2X)_X sup_(P(Y)∈R[Y]degP≤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 F̅(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+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+ε))), thus strengthening also previous work on the Fourier uniformity of the Liouville function.

Additional Information

© 2023 Department of Mathematics, Princeton University. 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 are grateful to the anonymous referees for their extremely careful reading of the paper and for numerous helpful comments and remarks that improved the presentation of this paper. We thank Amita Malik, Redmond McNamara and Peter Sarnak for helpful discussions.

Additional details

August 22, 2023
October 25, 2023