Suppose that $G$ is a compact Abelian topological group, $m$ is the Haar measure on $G$ and $f : G \to\mathbb{R}$ is a measurable function. Given $(n_k)$, a strictly monotone increasing sequence of integers we consider the nonconventional ergodic/Birkhoff averages
$$
M_N^\alpha f(x)=\frac1{N+A}\sum_{k=0}^N f(x+n_k\alpha).
$$
The $f$-rotation set is
$$
\Gamma_f = \{\alpha \in G : M_\alpha^N f (x)\textrm{ converges for }m\textrm{ a.e. }x\textrm{ as }N → ∞.\}
$$

We prove that if $G$ is a compact locally connected Abelian group and $f : G \to \mathbb{R}$ is a measurable function then from $m(\Gamma_f ) > 0$ it follows that $f ∈ L^1 (G)$.

A similar result is established for ordinary Birkhoff averages if $G = \mathbb{Z}_p$, the group of $p$-adic integers.

However, if the dual group, $G$ 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 $\Gamma_{f,b}$. This is a joint work with G. Keszthelyi.