A metric discrepancy result for the Hardy–Littlewood–Pólya sequences

2010

Acta Math Hung

Section: Introductionmentioning

confidence: 68%

Section: Idea Of the Proofmentioning

confidence: 96%

See 1 more Smart Citation

On the class of limits of lacunary trigonometric series

2010

Acta Math Hung

“…[4,11,21]), or if the sequence (n k ) k≥1 is of sub-lacunary growth (cf. [1,6,7,17,22]). There are only very few precise results which hold for general sequences (n k ) k≥1 without any growth conditions, except for the case n k = k, k ≥ 1, where the theory of continued fractions can be used to obtain precise estimates (cf.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

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

Probab. Theory Relat. Fields

Fukuyama

2011

Self Cite

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.

“…A comparison of (1.1) and (1.2) shows that the sequence {n k x} behaves like a sequence of independent random variables. However, as Fukuyama [6] and Fukuyama and Nakata [8] showed, the limsup in (1.1) is generally different from the constant 1/2 in (1.2), and it depends sensitively on the generating elements q 1 , . .…”

Section: Introductionmentioning

confidence: 99%

On permutations of Hardy-Littlewood-Pólya sequences

Trans. Amer. Math. Soc.

Berkes

Tichy

2011

Abstract. Let H = (q 1 , . . . , q r ) be a finite set of coprime integers and let n 1 , n 2 , . . . denote the multiplicative semigroup generated by H and arranged in increasing order. The distribution of such sequences has been studied intensively in number theory, and they have remarkable probabilistic and ergodic properties. In particular, the asymptotic properties of the sequence {n k x} are similar to those of independent, identically distributed random variables; here {·} denotes fractional part. In this paper we prove that under mild assumptions on the periodic function f , the sequence f (n k x) obeys the central limit theorem and the law of the iterated logarithm after any permutation of its terms. Note that the permutational invariance of the CLT and LIL generally fails for lacunary sequences f (m k x) even if (m k ) has Hadamard gaps. Our proof depends on recent deep results of Amoroso and Viada on Diophantine equations. We will also show that {n k x} satisfies a strong independence property ("interlaced mixing"), enabling one to determine the precise asymptotic behavior of permuted sums S N (σ) = N k=1 f (n σ(k) x).