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In this paper, we continue the study in our previous work Beurling dimenison and self-similar measures [J. Funct. Anal. 274 (2018) 2245–2264]. We analyze the Beurling dimension of Bessel sequences and frame spectra of self-affine measures on [Formula: see text]. Unlike the case for the self-similar measures, it is very difficult to obtain a general expression for the dimension of self-affine sets. This implies that the Hausdorff dimension may be not a good candidate that controls the Beurling dimension in general. Instead, we find that the pseudo Hausdorff dimension proposed by He and Lau [Math. Nachr. 281 (2008) 1142–1158] can be a good candidate that can control the Beurling dimension. With the help of the pseudo Hausdorff dimension, we obtain the upper bounds of the Beurling dimension of Bessel sequences. Under suitable condition, we give the lower bound of the Beurling dimension of frame spectra of self-affine measures on [Formula: see text]. Some examples are given to illustrate our theory.

Let [Formula: see text] be the unit matrix and [Formula: see text]. In this paper, we consider the self-similar measure [Formula: see text] on [Formula: see text] generated by the iterated function system [Formula: see text] where [Formula: see text]. We prove that there exists [Formula: see text] such that [Formula: see text] is an orthonormal basis for [Formula: see text] if and only if [Formula: see text] for some integer [Formula: see text].

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