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In this paper, we consider the non-spectral problem for the planar self-affine measures µ M,D generated by an expanding integer matrix M ∈ M 2 (Z) and a finite digitand gcd(det(M), p) = 1, then there exist at most p 2 mutually orthogonal exponential functions in L 2 (µ M,D ). In particular, if p is a prime, then the number p 2 is the best.

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The Cauchy transform of a measure has been used to study the analytic capacity and uniform rectifiability of subsets in C: Recently, Lund et al. (Experiment. Math. 7 (1998) 177) have initiated the study of such transform F of self-similar measure. In this and the forecoming papers (Starlikeness and the Cauchy transform of some self-similar measures, in preparation; The Cauchy transform on the Sierpinski gasket, in preparation), we study the analytic and geometric behavior as well as the fractal behavior of the transform F : The main concentration here is on the Laurent coefficients fa n g N n¼0 of F : We give asymptotic formulas for fa n g N n¼0 and for F ðkÞ ðzÞ near the support of m; hence the precise growth rates on ja n j and jF ðkÞ j are determined. These formulas are connected with some multiplicative periodic functions, which reflect the self-similarity of m and K: As a by-product, we also discover new identities of certain infinite products and series.

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