Coding discretizations of continuous functions

TL;DR

This paper investigates how discretizing continuous functions into finite codes reveals statistical and combinatorial properties, showing that all finite patterns can appear with any desired frequency in the limit.

Contribution

It introduces new results on the distribution of finite words in discretizations of continuous functions, demonstrating the universality of pattern frequencies in the limit.

Findings

Any finite word appears with any desired limit frequency in subsequence discretizations.

Discretizations reflect the variation of continuous functions at given precisions.

The study combines statistical and combinatorial analysis of coding sequences.

Abstract

We consider several coding discretizations of continuous functions which reflect their variation at some given precision. We study certain statistical and combinatorial properties of the sequence of finite words obtained by coding a typical continuous function when the diameter of the discretization tends to zero. Our main result is that any finite word appears on a subsequence discretization with any desired limit frequency.