On the Balog-Ruzsa Theorem in short intervals

TL;DR

This paper develops a short interval version of the Balog-Ruzsa theorem for bounds on exponential sums over r-free numbers and applies it to establish a lower bound for the sum involving the Möbius function.

Contribution

It introduces a short interval adaptation of the Balog-Ruzsa theorem and derives a new lower bound for the L1 norm of the Möbius function's exponential sum.

Findings

Established a lower bound of H^{1/6} for the L1 norm of the Möbius exponential sum.

Provided a short interval version of the Balog-Ruzsa theorem for r-free numbers.

Demonstrated the bound holds when H is significantly larger than N^{9/17 + ε}.

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