Higher uniformity of bounded multiplicative functions in short intervals on average


Let $\lambda$ denote the Liouville function. We show that, as $X \rightarrow \infty$,
\int^{2X}_X \sup_{\begin{smallmatrix} P(Y)\in\mathbb{R}[Y] \ \mathrm{deg}{P} \leq k\end{smallmatrix}}
\left| \sum_{x\leq n \leq x+H}
\lambda(n) e(-P(n))\right| dx=o (XH)
for all fixed $k$ and $X^{\theta} \leq H \leq X$ with $0 < \theta < 1$ fixed but arbitrarily small. Previously this was only established for $k \leq 1$. 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 $\overline{F}(g(n) \Gamma)$. By the inverse theory for the Gowers norms this implies the higher order asymptotic uniformity result
\int_{X}^{2X} \| \lambda \|_{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 $\lambda$ over short polynomial progressions $(n+P_1(m),\ldots, n+P_k(m))$, 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 $H\geq \exp((\log X)^{5/8+\varepsilon})$, thus strengthening also previous work on the Fourier uniformity of the Liouville function.


Kaisa Matomäki

Department of Mathematics and Statistics, University of Turku, 20014 Turku, Finland

Maksym Radziwiłł

Department of Mathematics, Caltech, 1200 E California Blvd, Pasadena, CA, 91125, USA

Terence Tao

Department of Mathematics, UCLA, 405 Hilgard Ave, Los Angeles, CA, 90095, USA

Joni Teräväinen

Mathematical Institute, University of Oxford, Woodstock Road, Oxford OX2 6GG, United Kingdom and Department of Mathematics, University of Turku, 20014 Turku, Finland

Tamar Ziegler

Einstein Institute of Mathematics, Givat Ram, The Hebrew University of Jerusalem, Edmond J. Safra Campus, Jerusalem 91904, Israel