### Abstract

It is known that the discrepancy D_{N}{ kx} of the sequence { kx} satisfies ND_{N}{ kx} = O((log N) (log log N) ^{1} ^{+} ^{ε}) a.e. for all ε> 0 , but not for ε= 0. For n_{k}= θ^{k}, θ> 1 we have ND_{N}{ n_{k}x} ≦ (Σ _{θ}+ ε) (2 Nlog log N) ^{1 / 2} a.e. for some 0 < Σ _{θ}< ∞ and N≧ N_{0} if ε> 0 , but not for ε< 0. In this paper we prove, extending results of Aistleitner–Larcher [6], that for any sufficiently smooth intermediate speed Ψ (N) between (log N) (log log N) ^{1} ^{+} ^{ε} and (Nlog log N) ^{1 / 2} and for any Σ > 0 , there exists a sequence { n_{k}} of positive integers such that ND_{N}{ n_{k}x} ≦ (Σ + ε) Ψ (N) eventually holds a.e. for ε> 0 , but not for ε< 0. We also consider a similar problem on the growth of trigonometric sums.

Language | English |
---|---|

Pages | 199-216 |

Number of pages | 18 |

Journal | Acta Mathematica Hungarica |

Volume | 151 |

Issue number | 1 |

DOIs | |

Status | Published - 1 Feb 2017 |

### Fingerprint

### Keywords

- discrepancy
- lacunary sequence
- law of the iterated logarithm

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Acta Mathematica Hungarica*,

*151*(1), 199-216. https://doi.org/10.1007/s10474-016-0658-2

**A metric discrepancy result with given speed.** / Berkes, I.; Fukuyama, K.; Nishimura, T.

Research output: Contribution to journal › Article › Research › peer-review

*Acta Mathematica Hungarica*, vol. 151, no. 1, pp. 199-216. https://doi.org/10.1007/s10474-016-0658-2

}

TY - JOUR

T1 - A metric discrepancy result with given speed

AU - Berkes, I.

AU - Fukuyama, K.

AU - Nishimura, T.

PY - 2017/2/1

Y1 - 2017/2/1

N2 - It is known that the discrepancy DN{ kx} of the sequence { kx} satisfies NDN{ kx} = O((log N) (log log N) 1 + ε) a.e. for all ε> 0 , but not for ε= 0. For nk= θk, θ> 1 we have NDN{ nkx} ≦ (Σ θ+ ε) (2 Nlog log N) 1 / 2 a.e. for some 0 < Σ θ< ∞ and N≧ N0 if ε> 0 , but not for ε< 0. In this paper we prove, extending results of Aistleitner–Larcher [6], that for any sufficiently smooth intermediate speed Ψ (N) between (log N) (log log N) 1 + ε and (Nlog log N) 1 / 2 and for any Σ > 0 , there exists a sequence { nk} of positive integers such that NDN{ nkx} ≦ (Σ + ε) Ψ (N) eventually holds a.e. for ε> 0 , but not for ε< 0. We also consider a similar problem on the growth of trigonometric sums.

AB - It is known that the discrepancy DN{ kx} of the sequence { kx} satisfies NDN{ kx} = O((log N) (log log N) 1 + ε) a.e. for all ε> 0 , but not for ε= 0. For nk= θk, θ> 1 we have NDN{ nkx} ≦ (Σ θ+ ε) (2 Nlog log N) 1 / 2 a.e. for some 0 < Σ θ< ∞ and N≧ N0 if ε> 0 , but not for ε< 0. In this paper we prove, extending results of Aistleitner–Larcher [6], that for any sufficiently smooth intermediate speed Ψ (N) between (log N) (log log N) 1 + ε and (Nlog log N) 1 / 2 and for any Σ > 0 , there exists a sequence { nk} of positive integers such that NDN{ nkx} ≦ (Σ + ε) Ψ (N) eventually holds a.e. for ε> 0 , but not for ε< 0. We also consider a similar problem on the growth of trigonometric sums.

KW - discrepancy

KW - lacunary sequence

KW - law of the iterated logarithm

UR - http://www.scopus.com/inward/record.url?scp=84992222208&partnerID=8YFLogxK

U2 - 10.1007/s10474-016-0658-2

DO - 10.1007/s10474-016-0658-2

M3 - Article

VL - 151

SP - 199

EP - 216

JO - Acta Mathematica Hungarica

T2 - Acta Mathematica Hungarica

JF - Acta Mathematica Hungarica

SN - 0236-5294

IS - 1

ER -