论文标题
具有任意维度的光谱措施
Spectral measures with arbitrary dimensions
论文作者
论文摘要
它已知[Dai and Sun，J。Funct。肛门。 268（2015），2464--2477]，存在任意豪斯多夫维度的光谱措施，并且自然而然地提出了一个问题，即在光谱测量的其他维度是否发生过类似现象。在本文中，我们首先获得了Assouad维度和较低维度的公式，用于一类维度措施，该措施是由An和He引入的[J. [J.功能。肛门。 266（2014），343--354]。基于这些结果，我们显示了具有任意收获尺寸的光谱度量的存在$ \ dim_a $和较低的尺寸$ \ dim_l $从$ 0 $到$ 1 $，包括非原子零维光谱度量和一维的单调频谱测度，并且两种值可能会偶联。实际上，还获得了更多，对于任何$ 0 \ leq t \ leq s \ leq s \ leq r \ leq u \ leq 1 $，存在一个频谱量度$μ$，这样，\ [\dim_lμ= t，\dim_hμ= s，\dim_hμ= $ \ dim_p $分别表示hausdorff尺寸和量子的包装维度$μ$。该结果更简单，灵活地改善了Dai和Sun的结果。
It is known [Dai and Sun, J. Funct. Anal. 268 (2015), 2464--2477] that there exist spectral measures with arbitrary Hausdorff dimensions, and it is natural to pose the question of whether similar phenomena occur for other dimensions of spectral measures. In this paper, we first obtain the formulae of Assouad dimension and of lower dimension for a class of Moran measures in dimension one that is introduced by An and He [J. Funct. Anal. 266 (2014), 343--354]. Based on these results, we show the existence of spectral measures with arbitrary Assound dimensions $\dim_A$ and lower dimensions $\dim_L$ ranging from $0$ to $1$, including non-atomic zero-dimensional spectral measures and one-dimensional singular spectral measures, and prove that the two values may coincide. In fact, more is obtained that for any $0 \leq t \leq s \leq r \leq u\leq 1$, there exists a spectral measure $μ$ such that \[\dim_L μ=t, \dim_H μ=s, \dim_Pμ=r~ \text{and} \dim_Aμ=u,\] where $\dim_H$ and $\dim_P$ denote the Hausdorff dimension and packing dimension of the measure $μ$, respectively. This result improves and generalizes the result of Dai and Sun more simply and flexibly.
