On the law of the iterated logarithm for trigonometric series with bounded gaps

Abstract

Let (n k) k≥1 be an increasing sequence of positive integers. Bobkov and Götze proved that if the distribution of cos 2πn 1 x + • • • + cos 2πn N x √ N (1) converges to a Gaussian distribution, then the value of the variance is bounded from above by 1/2 − lim sup k/(2n k). In particular it is impossible that for a sequence (n k) k≥1 with bounded gaps (i.e. n k+1 − n k ≤ c for some constant c) the distribution of (1) converges to a Gaussian distribution with variance σ 2 = 1/2 or larger. In this paper we show that the situation is considerably different in the case of the law of the iterated logarithm. We prove the existence of an increasing sequence of positive integers satisfying n k+1 − n k ≤ 2 such that lim sup N →∞ N k=1 cos 2πn k x √ 2N log log N = +∞ a.e.

Additive Energy and Irregularities of Distribution

“…Further examples of sequences (a n ) n≥1 with highest possible additive energy and the essentially maximal metric order of ND * N of ({a n α}) n≥1 can be deduced from the results in [1,2,7], where it may be necessary to modify the sequences constructed there by inserting long stretches of arithmetic progressions in order to maximize the additive energy. However, the examples given in these papers are randomly generated sequences and no explicit constructions are known, whereas the example in Theorem 4 above is fully explicit.…”

Statistical independence in mathematics–the key to a Gaussian law

“…The results presented in this final section are not restricted to central limit phenomena. Beyond the normal fluctuations one can also prove laws of the iterated logarithm for lacunary series and we refer the reader to the work of Erdős and Gál [21], Aistleitner and Fukuyama [4,5], Aistleitner, Berkes, and Tichy [2,3], and the references cited therein. The study of large deviation principles for lacunary sums has recently been initiated by Aistleitner, Gantert, Kabluchko, Prochno, and Ramanan in [6].…”

A metric discrepancy result with given speed

“…Typical examples are Ψ(N ) = N α (log N ) β (log log N ) γ where the parameters α, β, γ are chosen so that the order of growth of Ψ 2 (N ) is between the previous bounds. Note that the theorem does not cover Ψ(N ) = (N log log N ) 1/2 ; the existence of {n k } with (7) is already proved in [4] for 0 < Σ < ∞, and in [2] for Σ = ∞. See also [9,14].…”