Suppose that
$G$
is a compact Abelian topological group,
$m$
is the Haar measure on
$G$
and
$f:G\rightarrow \mathbb{R}$
is a measurable function. Given
$(n_{k})$
, a strictly monotone increasing sequence of integers, we consider the non-conventional ergodic/Birkhoff averages
$$\begin{eqnarray}M_{N}^{\unicode[STIX]{x1D6FC}}f(x)=\frac{1}{N+1}\mathop{\sum }_{k=0}^{N}f(x+n_{k}\unicode[STIX]{x1D6FC}).\end{eqnarray}$$
The
$f$
-rotation set is
$$\begin{eqnarray}\unicode[STIX]{x1D6E4}_{f}=\{\unicode[STIX]{x1D6FC}\in G:M_{N}^{\unicode[STIX]{x1D6FC}}f(x)\text{ converges for }m\text{ almost every }x\text{ as }N\rightarrow \infty \}.\end{eqnarray}$$
We prove that if
$G$
is a compact locally connected Abelian group and
$f:G\rightarrow \mathbb{R}$
is a measurable function then from
$m(\unicode[STIX]{x1D6E4}_{f})>0$
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,
$\widehat{G}$
, 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,
$f(x+n_{k}\unicode[STIX]{x1D6FC})/k$
,
$k=1,\ldots ,$
for almost every
$x$
for many
$\unicode[STIX]{x1D6FC}$
; hence, some of our theorems are stated by using instead of
$\unicode[STIX]{x1D6E4}_{f}$
slightly larger sets, denoted by
$\unicode[STIX]{x1D6E4}_{f,b}$
.