Exact limit theorems for restricted integer partitions

For a set of positive integers A, let p(A)(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, Erdos extended the classical Hardy-Ramanujan formula for p(n) by showing that A has density alpha if and only if log p(A)(n) similar to log p(alpha n). Nathanson asked if Erdos's theorem holds also with respect to A's lower density, namely, whether A has lower-density alpha if and only if log p(A)(n)/log p(alpha n) has lower limit 1. We answer this question negatively by constructing, for every alpha > 0, a set of integers A of lower density alpha, satisfying lim inf(n ->infinity) log p(A)(n)/log p(alpha n) >= (root 6/pi - o(alpha)(1)) log(1/alpha). We further show that the above bound is best possible (up to the o(alpha)(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. (C) 2022 Elsevier Inc. All rights reserved.