  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 M\"obius 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 $H \gg N^{\frac{9}{17} + \varepsilon}$.
