Let $$\lambda $$ λ denote the Liouville function. We show that as $$X \rightarrow \infty $$ X → ∞ , $$\begin{aligned} \int _{X}^{2X} \sup _{\alpha } \left| \sum _{x < n \le x + H} \lambda (n) e(-\alpha n) \right| dx = o ( X H) \end{aligned}$$ ∫ X 2 X sup α ∑ x < n ≤ x + H λ ( n ) e ( - α n ) d x = o ( X H ) for all $$H \ge X^{\theta }$$ H ≥ X θ with $$\theta > 0$$ θ > 0 fixed but arbitrarily small. Previously, this was only known for $$\theta > 5/8$$ θ > 5 / 8 . For smaller values of $$\theta $$ θ this is the first “non-trivial” case of local Fourier uniformity on average at this scale. We also obtain the analogous statement for (non-pretentious) 1-bounded multiplicative functions. We illustrate the strength of the result by obtaining cancellations in the sum of $$\lambda (n) \Lambda (n + h) \Lambda (n + 2h)$$ λ ( n ) Λ ( n + h ) Λ ( n + 2 h ) over the ranges $$h < X^{\theta }$$ h < X θ and $$n < X$$ n < X , and where $$\Lambda $$ Λ is the von Mangoldt function.

中文翻译：

---

平均短区间内有界乘法函数的傅里叶均匀性

让 $$\lambda $$ λ 表示 Liouville 函数。我们证明 $$X \rightarrow \infty $$ X → ∞ ， $$\begin{aligned} \int _{X}^{2X} \sup _{\alpha } \left| \sum _{x < n \le x + H} \lambda (n) e(-\alpha n) \right| dx = o ( XH) \end{aligned}$$ ∫ X 2 X sup α ∑ x < n ≤ x + H λ ( n ) e ( - α n ) dx = o ( XH ) 对于所有 $$H \ge X^{\theta }$$ H ≥ X θ $$\theta > 0$$ θ > 0 固定但任意小。以前，这仅适用于 $$\theta > 5/8$$ θ > 5 / 8 。对于 $$\theta $$ θ 的较小值，这是该尺度下平均局部傅立叶均匀性的第一个“非平凡”案例。我们还获得了（非自命不凡的）1 有界乘法函数的类似陈述。