In this paper, we show that the lower dimension is not invariant under quasi-Lipschitz mapping, and then we find an invariant named the quasi-lower dimension. We also compute the quasi-lower dimension of a class of sets defined by digit restrictions, and then give an example to distinguish the quasi-lower dimension and other dimensions.

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The homogeneous perfect sets introduced by Wen and Wu [Hausdorff dimension of homogeneous perfect sets, Acta. Math. Hungar. 107 (2005) 35–44] is an important class of Moran sets. In this paper, we obtain the Assouad dimension and Assouad spectrum formulas for homogeneous perfect set under suitable condition. In the proof an Assouad spectrum formula for a large class of fractal sets is established.