Number theoretic applications of a class of Cantor series fractal functions, II

2013/10/09 by Brian Li, Li, Brian, Bill Mance +1
Computer Science · Mathematics · #Advanced Mathematical Identities #Computability, Logic, AI Algorithms #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1310.2379

openalex publication_date 2013/10/09 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

It is well known that all numbers that are normal of order k in base b are also normal of all orders less than k. Another basic fact is that every real number is normal in base b if and only if it is simply normal in base b^k for all k. This may be interpreted to mean that a number is normal in base b if and only if all blocks of digits occur with the desired relative frequency along every infinite arithmetic progression. We reinterpret these theorems for the Q-Cantor series expansions and show that they are no longer true in a particularly strong way. The main theoretical result of this paper will be to reduce the problem of constructing normal numbers with certain pathological properties to the problem of solving a system of Diophantine relations.

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