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Representations for complex numbers with integer digits
1Institut für Mathematik, Universität für Bodenkultur, Gregor-Mendel-Straße 33, A-1180 Vienna, Austria.
Research in Number Theory
|November 16, 2020
Summary
We introduce the zeta-expansion, a complex number expansion method using integer digits with a complex base. This novel technique extends the beta-expansion framework and connects to torus maps and shift radix systems.
Area of Science:
- Number Theory
- Complex Analysis
- Dynamical Systems
Background:
- The beta-expansion is a well-established method for representing real numbers.
- Recent advancements include rotational beta-expansions for complex numbers.
- Understanding number expansions in complex domains is an active research area.
Purpose of the Study:
- To introduce and define the zeta-expansion as a complex generalization of the beta-expansion.
- To explore the properties and applications of the zeta-expansion.
- To establish connections between the zeta-expansion and other mathematical concepts.
Main Methods:
- Developing the theoretical framework for the zeta-expansion.
- Utilizing integer digits for expansion with a complex base.
- Analyzing the relationship with rotational beta-expansions.
- Investigating links to piecewise affine maps and shift radix systems.
Main Results:
- The zeta-expansion is presented as a complex number system.
- It successfully expands complex numbers using integer digits and a complex base.
- The expansion is shown to fit within the rotational beta-expansion framework.
- New relationships with piecewise affine maps of the torus and shift radix systems are established.
Conclusions:
- The zeta-expansion offers a novel approach to complex number representation.
- It unifies concepts from different areas of mathematics.
- Further research into its properties and applications is warranted.
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