论文标题
在Balog-Ruzsa定理上短时间
On the Balog-Ruzsa Theorem in short intervals
论文作者
论文摘要
在本文中，我们给出了Balog-Ruzsa定理的简短间隔版本，涉及$ r $ $ r $ free数字的$ L_1 $规范的界限。作为一个应用程序，我们为使用Möbius函数定义的指数总和的$ L_1 $规范提供了下限。也就是说，我们表明$$ \ int _ {\ mathbb t} \ left | \ sum_ {| n-n | <h}μ（n）e（nα）\ right | dα\ gg h^{\ frac {1} {6}} $$当$ h \ gg n^{\ frac {\ frac {9} {17} {17} + \ varepsilon} $。
In this paper we give a short interval version of the Balog-Ruzsa theorem concerning bounds for the $L_1$ norm of the exponential sum over $r$-free numbers. As an application, we give a lower bound for the $L_1$ norm of the exponential sum defined with the Möbius function. Namely we show that $$\int_{\mathbb T} \left|\sum_{|n-N|<H} μ(n)e(n α)\right| d α\gg H^{\frac{1}{6}}$$ when $H \gg N^{\frac{9}{17} + \varepsilon}$.
