The Discrepancy of (nkx)k=1∞ With Respect to Certain Probability Measures
| Autor: | Agamemnon Zafeiropoulos, Niclas Technau |
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| Rok vydání: | 2020 |
| Předmět: | |
| Zdroj: | The Quarterly Journal of Mathematics. 71:573-597 |
| ISSN: | 1464-3847 0033-5606 |
| DOI: | 10.1093/qmathj/haz058 |
| Popis: | 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. |
| Databáze: | OpenAIRE |
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