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On a problem of Konyagin. (English) Zbl 1167.11008

Let \((G,+)\) be an abelian group. For \(A\subseteq G\) and \(t\in G,\) let \(\nu(t)=|\{(a,b)\in A\times A~:~t=a+b\}|. \) Let \(\nu(A)=\min_{t\in A+A}\nu(t).\) Sergei V. Konyagin [see V. F. Lev, “Reconstructing integer sets from their representation functions”, Electron. J. Comb. 11, No. 1, Research paper R78, 6 p., electronic only (2004; Zbl 1068.11006)], asked the following question:
“Do there exist constants \(\varepsilon, C>0\) such that for every sufficiently large prime \(p\) and each set \(A\subseteq \mathbb Z/p\mathbb Z\) with \(|A|<\sqrt p,\) we have \(\nu(A)\leq C|A|^{1-\varepsilon}?\)”
In the paper under review the authors prove a result towards this direction. The main theorem states that there are positive constants \(C_1, C_2\) such that if \(A\subseteq \mathbb Z/p\mathbb Z\) verifies \(|A|<C_1p^{2^{-d-1}}\), \(d\) integer, \(d\geq 3\) then \[ \nu(A)\leq C_2|A|. \] More precise, but technically complicated, information connecting \(C_1,C_2\) and \(d\) is given.
Ingredients of the proof are Dirichlet’s approximation theorem and the following result of Plünnecke and Ruzsa:
“Let \(C,D\) be finite subsets of an abelian group. If \(|C+D|\leq K|D|,\) then for every \(k\geq1,\) \[ |kC|\leq K^k |D|. \]

MSC:

11B34 Representation functions
11B75 Other combinatorial number theory

Citations:

Zbl 1068.11006
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