## 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”.