The sum-product theorem in \(\mathbb Z_q\) with \(q\) arbitrary
For a non-empty subset \(A\) of \({\mathbb Z}_q={\mathbb Z}/q{\mathbb Z}\), define \[ A+A=\{a+b:\, a,b\in A\} \] and \[ A.A=\{a\cdot b:\, a,b\in A\}. \] When \(q=p\) is prime, \textit{J. Bourgain, N. Katz} and \textit{T. Tao} [Geom. Funct. Anal. 14, No. 1, 27--57 (2004; Zbl 1145.11306)] and \textit{J. Bourgain, A. A. Glibichuk} and \textit{S. V. Konyagin} [J. Lond. Math. Soc., II. Ser. 73, No. 2, 380--398 (2006; Zbl 1093.11057)] proved that for any fixed \(\varepsilon>0\), there exists \(\delta=\delta(\varepsilon)>0\) such that \[ \max\{|A+A|,|A.A|\}\geq |A|^{1+\delta} \] for any \(A\subseteq{\mathbb Z}_p\) with \(|A|\leq p^{1-\epsilon}\). In the present paper, the author discusses the lower bound of \(\max\{|A+A|,|A.A|\}\) for composite \(q\). For \(q'\mid q\), let \(\pi_{q'}:\;{\mathbb Z}_q\to{\mathbb Z}_{q'}\) be the residue map. For \(A\subseteq{\mathbb Z}_q\), \(0<\delta_1<1\) and \(0<\delta_2\leq 1\), suppose that (1) \(|A|<q^{1-\delta_1}\); (2) \(|\pi_{q_1}(A)|>q^{\delta_2}\) for all \(q_1\mid q\) with \(q_1>q^\varepsilon\), where \(\epsilon=\epsilon(\delta_1,\delta_2)\). Then the author proves that \[ |A+A|+|A.A|>q^{\delta_3}|A|, \] where \(\delta_3\) only depends on \(\delta_1\) and \(\delta_2\). Furthermore, using the above result, the author gives an upper bound for the exponential sum \[ \max_{\substack{ \xi\in{\mathbb Z}_q\\ (\xi,q)=1}} \bigg|\sum_{x_1\in A_1,\dots,x_k\in A_k}\exp(\xi x_1\cdots x_k/q)\bigg| \]
scientific article; zbMATH DE number 5508687
scientific article