p-Adic自相似弦的Minkowski可测性和精确分形管公式
Minkowski Measurability and Exact Fractal Tube Formulas for p-Adic Self-Similar Strings
摘要：
p-adic分形弦及其复维数的理论是由前两位作者在[17，18，19]中开发的，特别是在自相似的情况下，与第一和第三作者在[28]中开发的阿基米德（或实）对应物平行。利用作者在[20]中得到的p-adic分形弦的分形管公式，本文给出了p-adic自相似分形弦Lp的管状邻域的一个精确体积公式，它用复维数表示.复杂维度的周期性结构使人们能够获得一个非常具体的形式，作为结果的分形管公式。此外，我们推导并使用截断版本的分形管公式，以表明Lp是不是Minkowski可测的，并获得其平均Minkowski内容的明确表达。一般理论通过两个简单的例子来说明，3-adic Cantor弦和2-adic Fibonacci弦，它们是[28]中研究的真实Cantor和Fibonacci弦的非阿基米德类似物（在[17，18]中引入）。
英文摘要：
The theory of p-adic fractal strings and their complex dimensions was developed by the first two authors in [17, 18, 19], particularly in the self-similar case, in parallel with its archimedean (or real) counterpart developed by the first and third author in [28]. Using the fractal tube formula obtained by the authors for p-adic fractal strings in [20], we present here an exact volume formula for the tubular neighborhood of a p-adic self-similar fractal string Lp, expressed in terms of the underlying complex dimensions. The periodic structure of the complex dimensions allows one to obtain a very concrete form for the resulting fractal tube formula. Moreover, we derive and use a truncated version of this fractal tube formula in order to show that Lp is not Minkowski measurable and obtain an explicit expression for its average Minkowski content. The general theory is illustrated by two simple examples, the 3-adic Cantor string and the 2-adic Fibonacci strings, which are nonarchimedean analogs (introduced in [17, 18]) of the real Cantor and Fibonacci strings studied in [28].
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