Let K be a homogeneous self-similar set satisfying the strong separation condition. This paper is concerned with the quantitative recurrence properties of the natural map T : K → K induced by the shift. Let µ be the natural self-similar measure supported on K. For a positive function ϕ defined on N, we show that the µ-measure of the following set R(ϕ) := {x ∈ K : |T n x − x| < ϕ(n) for infinitely many n ∈ N} is null or full according to convergence or divergence of a certain series. Moreover, a similar dichotomy law holds for the general Hausdorff measure, which completes the metric theory of this set.2010 Mathematics Subject Classification. Primary 28A80; 28D05; Secondary 11K55.

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In this paper, we are concerned with the relationship among the lower Assouad type dimensions. For uniformly perfect sets in doubling metric spaces, we obtain a variational result between two different but closely related lower Assouad spectra. As an application, we show that the limit of the lower Assouad spectrum as θ tends to 1 equals to the quasi-lower Assouad dimension, which provides an equivalent definition to the latter. On the other hand, although the limit of the lower Assouad spectrum as θ tends to 0 exists, there exist uniformly perfect sets such that this limit is not equal to the lower box-counting dimension. Moreover, by the example of Cantor cut-out sets, we show that the new definition of quasi lower Assouad dimension is more accessible, and indicate that the lower Assouad dimension could be strictly smaller than the lower spectra and the quasi lower Assouad dimension.

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