Study on Multifractal and its Applications to the Digital Sum Problems

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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.