Exact limit theorems for restricted integer partitions (Q2166333)

For a set \(A\) of positive integers with gcd(\(A\))=1, 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. \textit{P. Erdős} [Ann. of Math. 43, 437--450 (1942; Zbl 0061.07905)] showed that \(A\) has density \(\alpha\) if and only if \(\log p_A(n)\sim \log p(\alpha n)\) as \(n\to \infty.\) \textit{M. B. Nathanson} [Acta Math. Hung. 87, 179--195 (2000; Zbl 0999.11062)] proved the following strengthening for a set \(A\) of integers with gcd(\(A\))=1 of lower density \(\alpha\) and upper density \(\beta:\) \[ \liminf_{n \to \infty} \frac {\log p_A(n)}{\log p(\alpha n)}\ge 1, \quad \limsup_{n \to \infty} \frac {\log p_A(n)}{\log p(\beta n)}\le 1. \] Nathanson [loc. cit.] asked if the above inequalities are in fact equalities. In the paper under review, the authors show that the answers to both questions are negative: For every \(0<\alpha<1,\) there is a set \(A\) of integers with gcd(\(A\))=1 of lower density \(\alpha\) satisfying \[ \liminf_{n \to \infty} \frac {\log p_A(n)}{\log p(\alpha n)}\ge (1-o_{\alpha}(1))\frac {\sqrt 6}{\pi} \log (1/\alpha). \] For every \(0<\beta<1,\) there is a set \(A\) of integers with gcd(\(A\))=1 of upper density \(\beta\) satisfying \[ \limsup_{n \to \infty} \frac {\log p_A(n)}{\log p(\beta n)}\le \frac {\sqrt 6 \log 2}{\pi}+o_{\beta}(1). \] The authors also show that these bounds are best possible.