Frequency of a Digit in the Representation of a Number and the Asymptotic Mean Value of the Digits

TL;DR

This paper investigates the relationship between digit frequency and the average digit value in ternary numbers, establishing conditions for the existence of the asymptotic mean and identifying numbers with unique properties.

Contribution

It introduces new conditions for the existence of the asymptotic mean of digits and identifies a dense set of numbers lacking digit frequency but having a well-defined mean.

Findings

Conditions for the existence of the asymptotic mean of digits

Existence of a dense set of numbers without digit frequency

Numbers with asymptotic mean but no digit frequency

Abstract

We study the relationship between the frequency of a ternary digit in a number and the asymptotic mean value of the digits. The conditions for the existence of the asymptotic mean of digits in a ternary number are established. We indicate an infinite everywhere dense set of numbers without frequency of digits but with the asymptotic mean of the digits.