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In this paper, using the feature of zero-point sets Z(...M,D), where µM,D is the Fourier transform of self-affine measures µM,D, we discuss the spectrality and non-spectrality of the collinear digit set. We give a method to deal with the spectrality and non-spectrality with collinear case. The results here provides some supportive evidence to the two related conjectures. [ABSTRACT FROM AUTHOR]

In this paper, using the feature of zero-point sets Z(µM,D), where µM,D is the Fourier transform of self-affine measures µM,D, we discuss the spectrality and non-spectrality of the collinear digit set. We give a method to deal with the spectrality and non-spectrality with collinear case. The results here provides some supportive evidence to the two related conjectures. [ABSTRACT FROM AUTHOR]

Abstract: In the paper, we focus on the connectedness of planar self-affine sets generated by an integer expanding matrix with and a collinear digit set , where and such that is linearly independent. We discuss the domain of the digit to determine the connectedness of . Especially, a complete characterization is obtained when we restrict to be an integer. Some results on the general case of are obtained as well. [Copyright &y& Elsevier]