Exact limit theorems for restricted integer partitions

Abstract

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, satisfyinglim infn→∞log⁡pA(n)log⁡p(αn)≥(6π−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.