Spectral measures with arbitrary dimensions

Abstract

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 $\mu$ such that \[\dim_L \mu=t, \dim_H \mu=s, \dim_P\mu=r~ \text{and} \dim_A\mu=u,\] where $\dim_H$ and $\dim_P$ denote the Hausdorff dimension and packing dimension of the measure $\mu$, respectively. This result improves and generalizes the result of Dai and Sun more simply and flexibly.