On Bourgain's bound for short exponential sums and squarefree numbers (Q2787076)

In the paper [Int. Math. Res. Not. 2015, No. 10, 2841--2855 (2015; Zbl 1372.11039)] \textit{J. Bourgain} proved a non-trivial bound for exponential sum NEWLINE\[NEWLINE\sum_{n\leq N\atop (n,q)=1}e\left(\frac{a \overline{n}^2}{ q}\right),NEWLINE\]NEWLINE where \(q>1\) is an integer and \(\overline{n}\) denotes the multiplicative inverse of \(n\pmod q\). The author applies this result in order to prove certain independence results related to the distribution of squarefree numbers in arithmetic progressions.NEWLINENEWLINELet \(X\geq 1\). Let \(a\) and \(q\) be coprime integers such that \(q\geq 2 \) and let NEWLINE\[NEWLINEE(X,q,a):=\sum_{{n\leq X\atop n\equiv a\pmod{q}}}\mu(n)^2-\frac{6}{ \pi^2}\prod_{p\mid q}\left(1-\frac{1}{ q^2}\right)^{-1}\frac{X}{q }.NEWLINE\]NEWLINE The author studies how \(E(X,q,a)\) correlates with \(E(X,q,\gamma_{r,s}(a))\) where \(\gamma_{r,s}(a)=ra+s\), \(r,s\in\mathbb{Z}\), \(r\neq 0\). For the correlation NEWLINE\[NEWLINEC[\gamma_{r,s}](X,q):=\sum_{{a\pmod q\atop a\neq 0,\gamma_{r,s}^{-1}(0)}}E(X,q,a)E(X,q,\gamma_{r,s}(a))NEWLINE\]NEWLINE he proves the following result.NEWLINENEWLINETheorem. There exist an absolute \(\delta>0\) such that for every \(\varepsilon>0\) and every integer \(r\neq 0,\) there exist \(C_{\varepsilon,r}\) such that NEWLINE\[NEWLINE|C[\gamma_{r,s}](X,q)|\leq C_{\varepsilon,r}\left(q^{1+\varepsilon}+X^{1/2}q^{1/2}\log^{-\delta}q+\frac{X^{5/3+\varepsilon}}{ q}+\left(\frac{X}{ q}\right)^2\right)NEWLINE\]NEWLINE uniformly for \(X\geq 2\), integers \(s\) and prime numbers \(q\leq X\) such that \(q\nmid rs\).