Study on Multifractal and its Applications to the Digital Sum Problems

本研究的目的是研究关于多项式测度的无穷多个差分方程组和函数方程组，这是一种典型的多重分形测度，并将这一研究应用于在p-进制数上展开的数字和问题。利用多项式测度对数字和问题进行了系统的研究。每个研究人员都积极地看待自己的研究对象，并做出了有益的贡献。我们列举了下面的结果。用多项式方法解决了常见的数字求和问题。给出了包含单位位数信息的幂和和与指数和的推广，并利用多项式测度给出了它们的显式公式。作为上述公式的应用，我们给出了子块在p-进展开式中出现的次数的一个显式公式。通过推广复系数多项式测度的分布函数，给出了与三的倍数中的二进制数字有关的另一个显式求和公式，这两个复杂性只是多项式测度推广的一个例子。在研究过程中，我们发现包含多项式度量的新的度量类对于数字和问题仍然是更有效的。我们应该为数字和问题的应用准备一个有效的理论。

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.