论文标题
平均间隔较高的有限乘函数的较高均匀性
Higher uniformity of bounded multiplicative functions in short intervals on average
论文作者
论文摘要
令$λ$表示Liouville功能。我们表明，作为$ x \ rightarrow \ infty $，$$ \ int_ {x}^{2x} {2x} \ sup _ {\ ordack {p（y）\ in \ mathbb {r} [y] \ sum_ {x \ leq n \ leq x + h}λ（n）e（-p（n））\ big | \ dx = o（x h）$（x h）$ 4（x h）$ 4，用于所有固定的$ k $和$ x^θ\ leq h \ leq h \ leq h \ leq x $，带有$ 0 <θ<1 <θ<1 $固定但固定小。以前，这仅以$ k \ leq 1 $建立。我们将此结果作为我们证明的（非先验）$ 1 $结合的乘法功能的相应语句的特殊情况。实际上，我们能够将多项式阶段$ e（-p（n））$替换为度$ k $ k $ nilsequences $ \ overline {f}（g（n）γ）$。通过逆向理论的戈尔斯规范，这意味着高阶渐近均匀性结果$ \ int_ {x}}^{2x} \ | | λ\ | _ {u^{k+1}（[x，x+h]）} \ dx = o（x）$$在$ h $的范围内。我们将此结果的应用介绍给Liouville序列中各种类型的模式。首先，我们表明，liouville函数的符号模式的数量是超级分类的，在萨尔纳克（Sarnak）的猜想上取得了进展，该sarnak涉及liouville序列具有正熵。其次，我们在短多项式进度$（n+p_1（m），\ ldots，n+p_k（m））$中以$λ$的平均值获得取消，在线性多项式的情况下，这会产生新的平均版本Chowla的猜想。实际上，我们能够证明我们在较大范围内的多项式阶段的结果$ h \ geq \ exp（（（\ log x）^{5/8+\ varepsilon}）$，从而在liouville函数的傅立叶均匀性上也加强了以前的工作。
Let $λ$ denote the Liouville function. We show that, as $X \rightarrow \infty$, $$\int_{X}^{2X} \sup_{\substack{P(Y)\in \mathbb{R}[Y]\\ deg(P)\leq k}} \Big | \sum_{x \leq n \leq x + H} λ(n) e(-P(n)) \Big |\ dx = o ( X H)$$ for all fixed $k$ and $X^θ \leq H \leq X$ with $0 < θ< 1$ 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 $\overline{F}(g(n) Γ)$. By the inverse theory for the Gowers norms this implies the higher order asymptotic uniformity result $$\int_{X}^{2X} \| λ\|_{U^{k+1}([x,x+H])}\ dx = o ( X )$$ 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 $(n+P_1(m),\ldots, n+P_k(m))$, 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.
