Frequencies of Patterns in Smooth Sequences Over the Alphabet $\{1,3\}$

2026-06-11

AI Summary

This study investigates the existence and well-definedness of asymptotic frequencies of arbitrary finite patterns in smooth sequences over the alphabet {1,3}. By introducing a type classification based on local structure, the authors construct a substitution system corresponding to the associated subshift and combine tools from ergodic theory and symbolic dynamics to establish criteria for minimality and unique ergodicity. The main contribution is the proof that the asymptotic frequency of every finite pattern in {1,3}-smooth sequences always exists and is uniquely determined by the derived type sequence, thereby providing a rigorous mathematical characterization of the statistical regularities inherent in smooth sequences.

Abstract

We provide an ergodic theory framework to study statistical properties of smooth sequences over the odd alphabet {1,3}. The arithmetic nature of this alphabet yields a partition of the subshift of smooth sequences based on their local structure, defining a notion of type for those sequences. We describe the substitutive structure of the smaller subshifts obtained by fixing the sequence of types of the successive derivatives of smooth sequences, from which we obtain the unique ergodicity of all these subshifts. A direct consequence is that the asymptotic frequency of any finite pattern in a smooth sequence over {1,3} is always well-defined and depends on its type sequence. Finally, we characterize the minimality of these subshifts.

Problem

• smooth sequences
• pattern frequencies
• subshifts
• ergodicity
• alphabet {1,3}

Innovation

• smooth sequences
• ergodic theory
• subshifts
• unique ergodicity
• substitutive structure