Logarithmic Representability of Integers as k-Sums

TL;DR

This paper demonstrates that a randomly chosen set in a specific probability space is highly likely to be a truncated logarithmic-representative k-basis for [n], extending classical results on additive bases.

Contribution

It establishes the probabilistic existence of truncated log-representative bases for [n], generalizing Erdős's classical results to finite settings.

Findings

Random sets are truncated log-representative bases with high probability.

Extends Erdős's classical infinite results to finite cases.

Provides probabilistic methods for additive basis construction.

Abstract

A set A=A_{k,n} in [n]\cup{0} is said to be an additive k-basis if each element in {0,1,...,kn} can be written as a k-sum of elements of A in at least one way. Seeking multiple representations as k-sums, and given any function phi(n), with lim(phi(n))=infinity, we say that A is a truncated phi(n)-representative k-basis for [n] if for each j in [alpha n, (k-alpha)n] the number of ways that j can be represented as a k-sum of elements of A_{k,n} is Theta(phi(n)). In this paper, we follow tradition and focus on the case phi(n)=log n, and show that a randomly selected set in an appropriate probability space is a truncated log-representative basis with probability that tends to one as n tends to infinity. This result is a finite version of a result proved by Erdos (1956) and extended by Erdos and Tetali (1990).