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The Solecki submeasures and densities on groups

We introduce the Solecki submeasure $σ(A)=\inf_F\sup_{x,y\in G}|F\cap xAy|/|F|$ and its left and right modifications on a group $G$, and study the interplay between the Solecki submeasure and the Haar measure on compact topological groups. Also we show that the right Solecki density on a countable amenable group coincides with the upper Banach density $d^*$ which allows us to generalize some fundamental results of Bogoliuboff, Folner, Cotlar and Ricabarra, Ellis and Keynes about difference sets and Jin, Beiglbock, Bergelson and Fish about the sumsets to the class of all amenable groups.