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. In particular, when r= 2, for H >= N9/17+epsilon, we have the lower bound resultintegral(T)vertical bar Sigma vertical bar(n-N vertical bar> H-1/3,and for H >= N18+29+(epsilon), we have the upper bound resultintegral(T)vertical bar Sigma vertical bar(n-N vertical bar> H-1/6 when H >= N9/17+epsilon.