R^d中自相似分形的Minkowski可测性
On the Minkowski Measurability of Self-Similar Fractals in R^d
摘要：
M. Lapidus和C. Pomerance（1990-1993）和K. J. Falconer（1995）证明了$\mathbb{R}$中的自相似分形是Minkowski可测的当且仅当它是非格型的。D. Gatzouras（1999）证明了$\mathbb{R}^d$中的自相似分形是Minkowski可测的，如果它是非格型的（尽管用他的方法实际计算内容是困难的），并证明了如果它是格型的，它不是Minkowski可测的。在温和的条件下，我们证明了这一猜想，并在非格的情况下，我们改进了他的结果在这个意义上，我们表示的内容的分形在剩余的相关$ζ$ -功能在Minkowski维。
英文摘要：
M. Lapidus and C. Pomerance (1990-1993) and K.J. Falconer (1995) proved that a self-similar fractal in $\mathbb{R}$ is Minkowski-measurable iff it is of non-lattice type. D. Gatzouras (1999) proved that a self-similar fractal in $\mathbb{R}^d$ is Minkowski measurable if it is of non-lattice type (though the actual computation of the content is intractable with his approach) and conjectured that it is not Minkowski measurable if it is of lattice type. Under mild conditions we prove this conjecture and in the non-lattice case we improve his result in the sense that we express the content of the fractal in terms of the residue of the associated $ζ$-function at the Minkowski-dimension.
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