Multi-Base Representations of Integers: Asymptotic Enumeration and Central Limit Theorems

TL;DR

This paper develops asymptotic formulas for counting multi-base integer representations and proves central limit theorems for digit-related statistics, enhancing understanding of their combinatorial and probabilistic properties.

Contribution

It introduces a general asymptotic enumeration formula and establishes central limit theorems for key digit statistics in multi-base representations.

Findings

Asymptotic formula for the number of multi-base representations of integers.

Central limit theorems for sum of digits and Hamming weight.

Results on digit occurrence distributions in random representations.

Abstract

In a multi-base representation of an integer (in contrast to, for example, the binary or decimal representation) the base (or radix) is replaced by products of powers of single bases. The resulting numeral system has desirable properties for fast arithmetic. It is usually redundant, which means that each integer can have multiple different digit expansions, so the natural question for the number of representations arises. In this paper, we provide a general asymptotic formula for the number of such multi-base representations of a positive integer $n$. Moreover, we prove central limit theorems for the sum of digits, the Hamming weight (number of non-zero digits, which is a measure of efficiency) and the occurrences of a fixed digits in a random representation.