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This article is cited in 33 scientific papers (total in 33 papers)
On additive shifts of multiplicative subgroups
I. V. Vyugina, I. D. Shkredovbc a Institute for Information Transmission Problems, Russian Academy of Sciences
b Steklov Mathematical Institute, Russian Academy of Sciences
c Laboratory of Discrete and Computational Geometry named after B. N. Delone of
P. G. Demidov Yaroslavl State University
Abstract:
It is proved that for an arbitrary subgroup $R\subseteq\mathbb Z/p\mathbb Z$ and any distinct nonzero elements $\mu_1,\dots,\mu_k$ we have
$$
\bigl|R\cap(R+\mu_1)\cap\dots\cap(R+\mu_k)\bigr|
\ll_k|R|^{{1}/{2}+\alpha_k}
$$
under the condition that $1\ll_k|R|\ll_kp^{1-\beta_k}$, where $\{\alpha_k\}$, $\{\beta_k\}$ are some
sequences of positive numbers such that $\alpha_k,\beta_k\to0$ as $k\to\infty$. Furthermore, it is shown that the inequality $|R\pm R|\gg|R|^{5/3}\log^{-1/2}|R|$ holds for any subgroup $R$ such that $|R|\ll p^{1/2}$.
Bibliography: 25 titles.
Keywords:
multiplicative subgroups, Stepanov's method, additive combinatorics.
Received: 22.02.2011
Citation:
I. V. Vyugin, I. D. Shkredov, “On additive shifts of multiplicative subgroups”, Mat. Sb., 203:6 (2012), 81–100; Sb. Math., 203:6 (2012), 844–863
Linking options:
https://www.mathnet.ru/eng/sm7857https://doi.org/10.1070/SM2012v203n06ABEH004245 https://www.mathnet.ru/eng/sm/v203/i6/p81
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Abstract page: | 885 | Russian version PDF: | 261 | English version PDF: | 11 | References: | 69 | First page: | 44 |
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