Kaisa Matomäki, Maksym Radziwiłł, Terence Tao, Joni Teräväinen, Tamar Ziegler

Ann. of Math. (2) 197 (2), 739-857, (March 2023) DOI: 10.4007/annals.2023.197.2.3
KEYWORDS: Chowla conjecture, Gowers uniformity, Liouville function, nilsequences, sign patterns, 11N37, 11B30, 37A44

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 $0 \lt \theta \lt 1$ 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 $H\geq \mathrm{exp}((\mathrm{log} X)^{5/8+\varepsilon})$, thus strengthening also previous work on the Fourier uniformity of the Liouville function.