On the Balog-Ruzsa Theorem in short intervals

Abstract

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 $${\int }_{\mathbb{T}}\left|\sum _{|n-N|<H}\mu (n)e(n\alpha )\right|d\alpha \gg H^{\frac{1}{6}}$$ when $H\gg N^{\frac{9}{17}+\epsilon }$.