On the asymptotic behavior of weakly lacunary series

Let $f$ be a measurable function satisfying $$f(x+1)=f(x), \qquad \int_0^1 f(x) dx=0, \qquad \textrm{Var} ~f < + \infty,$$ and let $(n_k)_{k\ge 1}$ be a sequence of integers satisfying $n_{k+1}/n_k \ge q >1$ $(k=1, 2, \ldots)$. By the classical theory of lacunary series, under suitable Diophantine conditions on $n_k$, $(f(n_kx))_{k\ge 1}$ satisfies the central limit theorem and the law of the iterated logarithm. These results extend for a class of subexponentially growing sequences $(n_k)_{k\ge 1}$ as well, but as Fukuyama (2009) showed, the behavior of $f(n_kx)$ is generally not permutation-invariant, e.g. a rearrangement of the sequence can ruin the CLT and LIL. In this paper we construct an infinite order Diophantine condition implying the permutation-invariant CLT and LIL without any growth conditions on $(n_k)_{k\ge 1}$ and show that the known finite order Diophantine conditions in the theory do not imply permutation-invariance even if $f(x)=\sin 2\pi x$ and $(n_k)_{k\ge 1}$ grows almost exponentially. Finally we prove that, in a suitable statistical sense, for almost all sequences $(n_k)_{k\ge 1}$ growing faster than polynomially, $(f(n_kx))_{k\ge 1}$ has permutation-invariant behavior.