Intersections of shifts of multiplicative subgroups
Let \(p\) be a prime and let \(\mathbb F_p=\mathbb Z/p\mathbb Z\) be a finite field of \(p\) elements. Applying a suitable generalization of \textit{S. A. Stepanov}'s method [Math. USSR, Izv. 3, 1103--1114 (1971; Zbl 0209.24401)] and using some results from \textit{F. K. Schmidt} [Math. Z. 45, 62--74 (1939; JFM 65.0115.02)] and \textit{A. Garcia} and \textit{J. F. Voloch} [Manuscr. Math. 59, 457--469 (1987; Zbl 0637.12015)], the authors obtain that Theorem: Let \(G_0, \ldots, G_k\subset \mathbb F_p^*, k\geq 1\) be multiplicative subgroups and let \(\mu_1, \ldots, \mu_k\in \mathbb F_p^*\) be distinct nonzero elements of \(\mathbb F_p\). If the following conditions hold {\parindent=0.7cm\begin{itemize}\item[(1)] \(|G_0|\cdot\ldots\cdot|G_s|<(s+1)^{-s-1/2}p^{s+1/2}\) for all \(s=1, \ldots, k\); \item[(2)] \[ \frac{1}{2}\big(\prod_{i=0}^{\ell}|G_i|\big)^{1/(2\ell+1)}<|G_m|<\frac{1}{2(\ell+3)}\big(\prod_{i=0}^{\ell}|G_i|\big)^{2/(2\ell+1)} \] for all \(\ell=0,\ldots, k\), \(m=0, \ldots, \ell\) and for the pair \(\ell=k-1, m=k\), \end{itemize}} then \[ |G_0\cap(G_1+\mu_1)\cap\ldots\cap(G_k+\mu_k)|\leq 16k(k+2)(|G_0|\cdot\ldots\cdot|G_k|)^{1/(2k+1)}\,. \] By the theorem in the particular case \(k=1\), the authors also get a result on the additive decomposability of multiplicative subgroups. Corollary: Let \(\epsilon\in (0,1]\) be a real number and let \(A, G\subset \mathbb F_p^*\) be multiplicative subgroups. Suppose \(|G|\geq C(\epsilon)\) is sufficiently large and \(B\subset \mathbb F_p^*\) is a nonempty subset. If \(|G\cap A|\leq |A|^{1-\epsilon}\), then \(G\) does not have a nontrivial representation of the form \(G=A+B\), where \(A+B=\{a+b\, |\, a\in A, b\in B\}\).
scientific article; zbMATH DE number 6682149
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