2013/02/07 by Anant P. Godbole, Samuel C. Gutekunst, Godbole, Anant P. +6
Computer Science · Mathematics · #60C05 #Analytic Number Theory Research #FOS: Mathematics #Limits and Structures in Graph Theory #Polynomial and algebraic computation #Probability (math.PR) #math.PR #msc:60C05
paper · pdf · doi:10.48550/arxiv.1302.1808
18 pages
arxiv created 2013/02/07 · openalex publication_date 2013/02/07 · arxiv updated 2013/02/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A set A=Ak,n in [n]∪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 Ak,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).