  Suppose that G is a compact Abelian topological group, m is the Haar measure
on G and f is a measurable function. Given (n_k), a strictly monotone
increasing sequence of integers we consider the nonconventional
ergodic/Birkhoff averages M_N^{\alpha}f(x). The f-rotation set is
Gamma_f={\alpha \in G: M_N^{\alpha} f(x) converges for m a.e. x as N\to \infty
.} We prove that if G is a compact locally connected Abelian group and f: G ->
R is a measurable function then from m(Gamma_f)>0 it follows that f \in L^1(G).
A similar result is established for ordinary Birkhoff averages if G=Z_{p}, the
group of p-adic integers. However, if the dual group, \hat{G} contains
"infinitely many multiple torsion" then such results do not hold if one
considers non-conventional Birkhoff averages along ergodic sequences. What
really matters in our results is the boundedness of the tail, f(x+n_{k}
{\alpha})/k, k=1,... for a.e. x for many \alpha, hence some of our theorems are
stated by using instead of Gamma_f slightly larger sets, denoted by
Gamma_{f,b}.
