부산대학교 도서관

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. In particular, when r  = 2, for |$H \geq N^{\frac{9}{17}+\epsilon}$|⁠ , we have the lower bound result $$ \int_{\mathbb T}\left|\sum_{|n-N| \lt H} \mu^2(n)e(n \alpha)\right| d \alpha \gg H^{\frac{1}{3}}, $$ and for |$H \geq N^{\frac{18}{29}+\epsilon}$|⁠ , we have the upper bound result $$ \int_{\mathbb T}\left|\sum_{|n-N| \lt H} \mu^2(n)e(n \alpha)\right| d \alpha \ll H^{\frac{1}{3}}.$$ As an application, we show that the L 1 norm of the exponential sum |$\sum_{|n-N| \lt H} \mu(n)e(n \alpha)$| has the lower bound |$\gg H^{\frac{1}{6}}$| when |$H \geq N^{\frac{9}{17} + \varepsilon}$|⁠. [ABSTRACT FROM AUTHOR]