Study on Multifractal and its Applications to the Digital Sum Problems

本课题的研究目的是研究一个关于多项式测度的无穷多个差分方程和泛函方程系统，多项式测度是一种典型的多重分形测度，并将此研究应用于p进数展开的数字和问题。利用多项测度系统地研究了数字和问题。每位研究者都积极地考虑自己的课题，并做出了有益的贡献。我们将结果列举如下。常用的数字和问题采用多项测度法解决。介绍了幂和指数和的一种推广方法，并利用多项测度给出了幂和指数和的显式表达式。作为上述公式的一个应用，我们给出了p进展开式中子块出现次数的显式公式。通过推广复系数多项测度的分布函数，给出了Coquet的与3的倍数有关的二进制数求和公式的另一个显式公式。上述复杂性只是多项式测度的一个推广实例。在研究过程中，我们发现包含多项测度的新测度类对于数字和问题仍然更有效。我们应该为数字和问题的应用准备一个有效的这些措施的理论。

The purpose of this research project is to study a system of infinitely many difference equations and functional equation with respect to the multinomial measure, which is a typical multifractal measure, and to apply this study to the digital sum problems expanded in the p-adic number. We investigated the digital sum problems systematic by using the multinomial measure. Each investigator considered his subject positively and made usefulcontribution. We enumerate the results in the following.1. Usual digital sum problems were solved by using the multinomial measures.2. We introduced a generalization of the power and the exponential sums, which contained information per digit, and gave explicit formulas of them by using the multinomial measure.3. As an application of the formula obtained above, we gave an explicit formula of the number of occurrences of subblock in the p-adic expansion.4. By generalizing the distribution function of multinomial measure with complex coefficients, we gave another explicit formula of Coquet's summation formula related to the binary digits in the multiple of three.Above complexity is only one instance of a generalization of the multinomial measure. In the process of this study, we found that new classes of measures, which contain the multinomial measures, are still more effective for the digital sum problems. We should prepare an effective theory of these measures for applications of the digital sum problems.