Acta mathematica scientia,Series A ›› 2022, Vol. 42 ›› Issue (2): 338-352.

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On the Optimal Voronoi Partitions for Moran Measures on ${\Bbb R} ^{1}$ with Respect to the Geometric Mean Error

Yi Cao()   

  1. School of Mathematics and Physics, Jiangsu University of Technology, Jiangsu Changzhou 213001
  • Received:2021-08-02 Online:2022-04-26 Published:2022-04-18
  • Supported by:
    the NSFC(11571144)

Abstract:

Let $ E $ be a Moran set on $ {{\Bbb R}} ^{1} $ associated with a bounded closed interval $ J $ and two sequences $ (n_{k})^{\propto}_{k=1} $ and $ {\cal C}_{k}=((c_{k, j})_{j=1}^{n_{k}})_{k\geq 1} $ of numbers. Let $ \mu $ be the Moran measure on $ E $ determined by a sequence $ ({\cal P}_{k})_{k\geq1} $ of positive probability vectors. For every $ n\geq 1 $, let $ C_{n}(\mu) $ denote the collection of all the $ n $-optimal sets for $ \mu $ with respect to the geometric mean error; let $ \alpha_n\in C_n(\mu) $ and $ \{P_{a}(\alpha_{n})\}_{a\in\alpha_{n}} $ be an arbitrary Voronoi partition with respect to $ \alpha_n $. We prove that For each $ a\in\alpha_{n} $, we show that the set $ P_{a}(\alpha_{n}) $ contains a closed interval of radius $ d_{2}|P_{a}(\alpha_{n})\cap E| $ which is centered at $ a $, where $ d_{2} $ is a constant and $ |B| $ denotes the diameter of a set $ B\subset{{\Bbb R}} ^1 $. Let $ e_n(\mu) $ denote the $ n $th geometric mean error for $ \mu $ and $ \hat{e}_n(\mu):=\log e_n(\mu) $. We show that $ \hat{e}_n(\mu)-\hat{e}_{n+1}(\mu)\asymp n^{-1} $.

Key words: Geometric mean error, Optimal Voronoi partition, Moran measure

CLC Number: 

  • O174.1
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