2021/04/06 by Asaf Cohen Antonir, A. Shapira, Antonir, Asaf Cohen +1
Mathematics · #Analytic Number Theory Research #Combinatorics (math.CO) #FOS: Mathematics #Graph theory and applications #Limits and Structures in Graph Theory #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2104.02692
openalex publication_date 2021/04/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a set of positive integers A, let pA(n) denote the number of ways to write n as a sum of integers from A, and let p(n) denote the usual partition function. In the early 40s, Erdős extended the classical Hardy--Ramanujan formula for p(n) by showing that A has density α if and only if log pA(n) ∼ log p(αn). Nathanson asked if Erdős's theorem holds also with respect to A's lower density, namely, whether A has lower-density α if and only if log pA(n) / log p(αn) has lower limit 1. We answer this question negatively by constructing, for every α> 0, a set of integers A of lower density α, satisfying \liminfn → ∞ (log pA(n))/(log p(αn)) ≥ (\frac√(6)π-oα(1))log(1/α) . We further show that the above bound is best possible (up to the oα(1) term), thus determining the exact extremal relation between the lower density of a set of integers and the lower limit of its partition function. We also prove an analogous theorem with respect to the upper density of a set of integers, answering another question of Nathanson.