The covering number of the difference sets in partitions of G-spaces and groups

We prove that for every finite partition $G=A_1\cup\dots\cup A_n$ of a group $G$ either $cov(A_iA_i^{-1})\le n$ for all cells $A_i$ or else $cov(A_iA_i^{-1}A_i)<n$ for some cell $A_i$ of the partition. Here $cov(A)=\min\{|F|:F\subset G,\;G=FA\}$ is the covering number of $A$ in $G$. A similar result is proved also of partitions of $G$-spaces. This gives two partial answers to a problem of Protasov posed in 1995.