TY - JOUR

T1 - The Discrepancy of (nkx)k=1 with Respect to Certain Probability Measures

AU - Technau, Niclas

AU - Zafeiropoulos, Agamemnon

N1 - Publisher Copyright:
© 2020 The Author(s) 2020. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: journals. permissions@oup.com.

PY - 2020/6/15

Y1 - 2020/6/15

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.

UR - http://www.scopus.com/inward/record.url?scp=85088978531&partnerID=8YFLogxK

U2 - 10.1093/qmathj/haz058

DO - 10.1093/qmathj/haz058

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AN - SCOPUS:85088978531

SN - 0033-5606

VL - 71

SP - 573

EP - 597

JO - Quarterly Journal of Mathematics

JF - Quarterly Journal of Mathematics

IS - 2

ER -