We investigate the almost everywhere convergence of
∑
c
n
f
(
n
x
)
\sum c_{n} f(nx)
, where
f
f
is a measurable function satisfying
f
(
x
+
1
)
=
f
(
x
)
,
∫
0
1
f
(
x
)
d
x
=
0.
\begin{equation*} f(x+1) = f(x), \qquad \int _{0}^{1} f(x) \, dx =0.\end{equation*}
By a known criterion, if
f
f
satisfies the above conditions and belongs to the Lip
α
\alpha
class for some
α
>
1
/
2
\alpha > 1/2
, then
∑
c
n
f
(
n
x
)
\sum c_{n} f(nx)
is a.e. convergent provided
∑
c
n
2
>
+
∞
\sum c_{n}^{2} > +\infty
. Using probabilistic methods, we prove that the above result is best possible; in fact there exist Lip 1/2 functions
f
f
and almost exponentially growing sequences
(
n
k
)
(n_{k})
such that
∑
c
k
f
(
n
k
x
)
\sum c_{k} f(n_{k} x)
is a.e. divergent for some
(
c
k
)
(c_{k})
with
∑
c
k
2
>
+
∞
\sum c_{k}^{2} > +\infty
. For functions
f
f
with Fourier series having a special structure, we also give necessary and sufficient convergence criteria. Finally we prove analogous results for the law of the iterated logarithm.