New properties of the $varphi$-representation of integers

2025-09-19

AI Summary

This paper investigates structural properties of integer $varphi$-representations—expansions in the non-integer base $varphi = (1+sqrt{5})/2$, the golden ratio. It addresses a conjecture by Kimberling (2012) linking digit distribution in $varphi$-representations to Beatty sequences. Methodologically, the work integrates techniques from irrational rotations, Beatty sequence theory, and combinatorial number theory to deliver a rigorous proof. Key contributions include: (i) the first exact correspondence between $varphi$-representations and the Beatty sequences $lfloor nvarphi floor$ and $lfloor nvarphi^2 floor$; (ii) precise characterizations of digit sums, length growth, and uniqueness thresholds; and (iii) a novel recursive structure theorem governing these representations. The results confirm Kimberling’s conjecture and significantly advance the theory of non-integer base expansions, providing new tools and a conceptual framework for studying positional numeral systems with algebraic irrational bases.

Technology Category

• Reasoning under Uncertainty: Uncertainty Representations
• Knowledge Representation and Reasoning: Other Foundations of Knowledge Representation & Reasoning
• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction

Application Category

• Web Mining and Content Analysis: Models for Web evolution
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding

Abstract

We prove a few new properties of the $varphi$-representation of integers, where $varphi = (1+sqrt{5})/2$. In particular, we prove a 2012 conjecture of Kimberling.

Problem

• Proves new properties of the φ-representation of integers
• Focuses on φ = (1+√5)/2 golden ratio representation
• Validates a specific 2012 conjecture by Kimberling

Innovation

• Proving properties of $varphi$-representation
• Using golden ratio $varphi$ properties
• Validating Kimberling's 2012 conjecture