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Roth's theorem on arithmetic progression states that a subset A of the natural numbers of positive upper density contains an arithmetic progression of length $3$, that is, the equation $x+z=2y$ has a solution in $A$. Likewise, one can consider a similar statement for finite groups, by considering the normalized counting measure, and ask whether the equation $x\cdot z=y\cdot y$ has a solution in a set of a fixed density. In this talk, I will explain how to use the model-theoretic notion of connected component, as well as of stability, to obtain Roth-like statements for suitable definably amenable groups. This is joint work with A. Martin-Pizarro (Freiburg).