Rudin-Shapiro Sums Via Automata Theory and Logic

Area of Science:

• Number Theory
• Theoretical Computer Science
• Mathematical Logic

Background:

• The Rudin-Shapiro sequence is a fundamental object in number theory and dynamical systems.
• Classical results by Brillhart and Morton established key properties of Rudin-Shapiro sums.
• Existing proofs often rely on specialized techniques, lacking a unified approach.

Purpose of the Study:

• To present a unified framework for analyzing Rudin-Shapiro sums.
• To derive classical results by Brillhart and Morton using this new framework.
• To demonstrate the framework's utility in discovering novel results.

Main Methods:

• Employing tools from logic and automata theory.
• Developing a unified theoretical framework.
• Applying the framework to analyze number-theoretic sums.

Main Results:

• A unified derivation of numerous classical results concerning Rudin-Shapiro sums.
• The framework provides simplified proofs for existing theorems.
• The methodology facilitates the discovery of new properties and results.

Conclusions:

• Logic and automata theory offer a powerful and unified approach to studying Rudin-Shapiro sums.
• The presented framework simplifies existing proofs and opens avenues for new research.
• This work bridges theoretical computer science and number theory.