On the Balog–Ruzsa theorem in short intervals

Abstract

Abstract In this paper we give a short interval version of the Balog–Ruzsa theorem concerning bounds for the L1 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 L1 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 $H \geq N^{\frac{9}{17} + \varepsilon}$.