On the asymptotic behavior of weakly lacunary series

Abstract

Abstract. Let f be a measurable function satisfyingand let (n k ) k≥1 be a sequence of integers satisfying n k+1 /n k ≥ q > 1 (k = 1, 2, . . .). By the classical theory of lacunary series, under suitable Diophantine conditions on n k , (f (n k x)) k≥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≥1 as well, but as Fukuyama showed, the behavior of f (n k x) 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≥1 and show that the known finite order Diophantine conditions in the theory do not imply permutation-invariance even if f (x) = sin 2πx and (n k ) k≥1 grows almost exponentially. Finally, we prove that in a suitable statistical sense, for almost all sequences (n k ) k≥1 growing faster than polynomially, (f (n k x)) k≥1 has permutation-invariant behavior.

Diophantine approximations and directional discrepancy of rotated lattices

“…Going one level back we write C q k = ν J ν as a union of closed dyadic intervals J ν of length 2 −L k . We observe that E q,j consists of open intervals of length 3δ(q) whose centers are equally spaced with step 1 q and A q,j inherits roughly the same structure. If q ≥ q k+1 , then…”

“…Consider a weakly lacunary sequence Ω = {ω k } ⊂ [0, 1] converging to 0 satisfying 0 ≤ ω k+1 ω k ≤ 1 − ck a , for some for −1 < a ≤ 0, see e.g. [1]. We then have F (x) ≈ x • log 1/(1+a) (1/x) and and…”

“…Ω = 2 −2 n . In this case, F (x) ≈ x • log log1 x , hence F −1 (x) ≈x log log(1/x) . As shown in (3.28), Theorem 2.1 guarantees the existence of α such that…”

On the Uniform Theory of Lacunary Series

“…The purpose of the present chapter is to provide a complete solution of the permutation-invariance of such results. For the proofs we refer to Aistleitner, Berkes and Tichy [2], [3], [5], [6]. Note that for the unpermuted CLT and LIL we need much weaker gap conditions than (1.1).…”

On permutational invariance of the metric discrepancy results

“…Aistleitner-Berkes-Tichy [1][2][3] studied the effect of permutation on metric discrepancy results. As a corollary, they derived the beautiful result below.…”