Abstract Suppose that q = 2 m + 1 ≥ 5 . Let F be the Cauchy transform of the self-similar measure μ = 1 q ∑ j = 0 q − 1 μ ∘ S j where S j ( z ) = e 2 j π i / q + ρ ( z − e 2 j π i / q ) with ρ ∈ ( 0 , 1 ) . Let K be the attractor of { S j } j = 0 q − 1 and R q = dist ( 0 , K ) . The Laurent coefficients of F in | z | > 1 was studied in [18] and [2] . In this paper, we study the asymptotic formula of the Taylor coefficients { b q k − 1 } k = 1 ∞ of F in | z | R q . We prove that the supremum of β satisfying { ( q k ) β R q q k b q k − 1 } k = 1 ∞ ∈ l ∞ is the Hausdorff dimension α of K, and that the accumulation points set of { ( q k ) α R q q k b q k − 1 } k = 1 ∞ is a union of some non-degenerate line segments.

中文翻译：

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一些 Hausdorff 测度的柯西变换的泰勒系数估计 (I)

摘要 假设 q = 2 m + 1 ≥ 5 。设 F 为自相似测度的柯西变换 μ = 1 q ∑ j = 0 q − 1 μ ∘ S j 其中 S j ( z ) = e 2 j π i / q + ρ ( z − e 2 j π i / q ) 与 ρ ∈ ( 0 , 1 ) 。设 K 为 { S j } j = 0 q − 1 和 R q = dist ( 0 , K ) 的吸引子。F in | 的洛朗系数 | | > 1 在 [18] 和 [2] 中进行了研究。在本文中，我们研究了 F in | 的泰勒系数 { bqk − 1 } k = 1 ∞ 的渐近公式。| | Rq。我们证明满足 { ( qk ) β R qqkbqk − 1 } k = 1 ∞ ∈ l ∞ 的 β 的上限值是 K 的 Hausdorff 维数 α，并且证明 { ( qk ) α R qqkbqk − 1 } 的累积点集k = 1 ∞ 是一些非退化线段的并集。