Exact Limit Theorems for Restricted Integer Partitions

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 lim inf n→∞ log pA(n) log p(αn) ≥ (√ 6 π − oα(1) )