Generalized golden ratios of ternary alphabets

TL;DR

This paper investigates the critical bases for unique expansions in ternary alphabets, revealing their fractal structure and extending known results from binary cases to three-letter systems.

Contribution

It determines the critical bases for all three-letter alphabets and analyzes their fractal properties, advancing understanding of noninteger base expansions.

Findings

Identified critical bases for ternary alphabets

Established the fractal nature of these bases

Extended binary alphabet results to ternary systems

Abstract

Expansions in noninteger bases often appear in number theory and probability theory, and they are closely connected to ergodic theory, measure theory and topology. For two-letter alphabets the golden ratio plays a special role: in smaller bases only trivial expansions are unique, whereas in greater bases there exist nontrivial unique expansions. In this paper we determine the corresponding critical bases for all three-letter alphabets and we establish the fractal nature of these bases in function of the alphabets.