Geodesic
On additive shifts of multiplicative subgroups
Sbornik. Mathematics, Tome 203 (2012) no. 6, pp. 844-863 Cet article a éte moissonné depuis la source Math-Net.Ru

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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. 
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I. V. Vyugin; I. D. Shkredov. On additive shifts of multiplicative subgroups. Sbornik. Mathematics, Tome 203 (2012) no. 6, pp. 844-863. http://geodesic.mathdoc.fr/item/SM_2012_203_6_a3/

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