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Multiple set addition in \(\mathbb Z_p\). (English) Zbl 1089.11009

Summary: It is shown that there exists an absolute constant \(H\) such that for every \(h>H\), every prime \(p\), and every set \(A\subseteq \mathbb Z_p\) such that \(10\leq |A|\leq p(\ln h)^{1/2}/(9h^{9/4})\) and \(|hA|\leq {h^{3/2}|A|}/{(8(\ln h)^{1/2})}\), the set \(A\) is contained in an arithmetic progression modulo \(p\) of cardinality \(\max_{1\leq j\leq h-1} \frac{|hA|-P_j(|A|)}{h-j}+1\), where \(P_j(n)=\frac{(j+1)j}{2}n-j^2+1\). This result can be viewed as a generalization of Freiman’s “2.4-theorem”.

MSC:

11B13 Additive bases, including sumsets

Keywords:

Freiman theorem
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