A nilpotent Roth theorem

Let T and S be invertible measure preserving transformations of a probability measure space (X, B, mu). We prove that if the group generated by T and S is nilpotent, then lim(N-->infinity) 1/N Sigma(n=1)(N) u(T(n)x)v(S(n)x) exists in L-2-norm for any u, v epsilon L-infinity(X, B, mu). We also show that for A epsilon B with mu(A) > 0 one has lim(N-->infinity) 1/N Sigma(n=1)(N) mu(A boolean AND T-n A boolean AND S-n A) > 0. By the way of contrast, we bring examples showing that if measure preserving transformations T, S generate a solvable group, then (i) the above limits do not have to exist; (ii) the double recurrence property fails, that is, for some A epsilon B, mu(A) > 0, one may have mu(A boolean AND T-n A boolean AND S-n A) = 0 for all n epsilon N. Finally, we show that when T and S generate a nilpotent group of class less than or equal to c, lim(N-->infinity) 1/N Sigma(n=1)(N) u(T(n)x)v(S(n)x) = integral udmu integral vdmu in L-2(X) for all u, v epsilon L-infinity(X) if and only if T x S is ergodic on X x X and the group generated by T-1 S, (T-2S2),...,(T-cSc) acts ergodically on X.