On the asymptotic behavior of weakly lacunary series

Let f be a measurable function satisfying f(x + 1) = f(x),| 1 0 f(x) dx = 0, Var[0,1]f < +∞,and let (nk)k≥1 be a sequence of integers satisfying nk+1/nk ≥ q > 1 (k = 1, 2, ⋯). By the classical theory of lacunary series, under suitable Diophantine conditions on nk, (f(nkx))k≥1 satisfies the central limit theorem and the law of the iterated logarithm. These results extend for a class of subexponentially growing sequences (nk)k≥1 as well, but as Fukuyama showed, the behavior of f(nkx) 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 (nk)k≥1 and show that the known finite order Diophantine conditions in the theory do not imply permutation-invariance even if f(x) = sin2πx and (nk)k≤1 grows almost exponentially. Finally, we prove that in a suitable statistical sense, for almost all sequences (nk)k≤1 growing faster than polynomially, (f(nkx))κ=1 has permutation-invariant behavior.