The discrepancy of $(n_kx)_{k=1}^{\infty}$ with respect to certain probability measures

Agamemnon Zafeiropoulos, Niclas Technau

Research output: Working paperResearchpeer-review

Abstract

Let $(n_k)_{k=1}^{\infty}$ be a lacunary sequence of integers. We show that if $\mu$ is a probability measure on $[0,1)$ such that $|\widehat{\mu}(t)|\leq c|t|^{-\eta}$, then for $\mu$-almost all $x$, the discrepancy $D_N(n_kx)$ satisfies \begin{equation*} \frac{1}{4} \leq \limsup_{N\to\infty}\frac{N D_N(n_kx)}{\sqrt{N\log\log N}} \leq C \end{equation*} for some constant $C>0$. This proves a conjecture of Haynes, Jensen and Kristensen and allows an improvement on their previous result relevant to an inhomogeneous version of the Littlewood Conjecture.
Original languageEnglish
Number of pages26
Publication statusPublished - 2018

Fingerprint

Probability Measure
Discrepancy
Lacunary Sequence
Integer

Cite this

The discrepancy of $(n_kx)_{k=1}^{\infty}$ with respect to certain probability measures. / Zafeiropoulos, Agamemnon; Technau, Niclas.

2018.

Research output: Working paperResearchpeer-review

@techreport{45642dce8fdb4b48a49deb5aa184bf43,
title = "The discrepancy of $(n_kx)_{k=1}^{\infty}$ with respect to certain probability measures",
abstract = "Let $(n_k)_{k=1}^{\infty}$ be a lacunary sequence of integers. We show that if $\mu$ is a probability measure on $[0,1)$ such that $|\widehat{\mu}(t)|\leq c|t|^{-\eta}$, then for $\mu$-almost all $x$, the discrepancy $D_N(n_kx)$ satisfies \begin{equation*} \frac{1}{4} \leq \limsup_{N\to\infty}\frac{N D_N(n_kx)}{\sqrt{N\log\log N}} \leq C \end{equation*} for some constant $C>0$. This proves a conjecture of Haynes, Jensen and Kristensen and allows an improvement on their previous result relevant to an inhomogeneous version of the Littlewood Conjecture.",
author = "Agamemnon Zafeiropoulos and Niclas Technau",
year = "2018",
language = "English",
type = "WorkingPaper",

}

TY - UNPB

T1 - The discrepancy of $(n_kx)_{k=1}^{\infty}$ with respect to certain probability measures

AU - Zafeiropoulos, Agamemnon

AU - Technau, Niclas

PY - 2018

Y1 - 2018

N2 - Let $(n_k)_{k=1}^{\infty}$ be a lacunary sequence of integers. We show that if $\mu$ is a probability measure on $[0,1)$ such that $|\widehat{\mu}(t)|\leq c|t|^{-\eta}$, then for $\mu$-almost all $x$, the discrepancy $D_N(n_kx)$ satisfies \begin{equation*} \frac{1}{4} \leq \limsup_{N\to\infty}\frac{N D_N(n_kx)}{\sqrt{N\log\log N}} \leq C \end{equation*} for some constant $C>0$. This proves a conjecture of Haynes, Jensen and Kristensen and allows an improvement on their previous result relevant to an inhomogeneous version of the Littlewood Conjecture.

AB - Let $(n_k)_{k=1}^{\infty}$ be a lacunary sequence of integers. We show that if $\mu$ is a probability measure on $[0,1)$ such that $|\widehat{\mu}(t)|\leq c|t|^{-\eta}$, then for $\mu$-almost all $x$, the discrepancy $D_N(n_kx)$ satisfies \begin{equation*} \frac{1}{4} \leq \limsup_{N\to\infty}\frac{N D_N(n_kx)}{\sqrt{N\log\log N}} \leq C \end{equation*} for some constant $C>0$. This proves a conjecture of Haynes, Jensen and Kristensen and allows an improvement on their previous result relevant to an inhomogeneous version of the Littlewood Conjecture.

M3 - Working paper

BT - The discrepancy of $(n_kx)_{k=1}^{\infty}$ with respect to certain probability measures

ER -