Exact Limit Theorems for Restricted Integer Partitions

TL;DR

This paper investigates the relationship between the lower and upper densities of integer sets and the asymptotic behavior of their restricted partition functions, providing exact bounds and counterexamples to previous conjectures.

Contribution

It constructs sets of integers with prescribed densities that demonstrate the limits of Erdős's extension of Hardy-Ramanujan formulas, and establishes optimal bounds for these relations.

Findings

Counterexamples to Nathanson's question about lower density

Exact bounds relating set density to partition function growth

Extension of results to upper density cases

Abstract

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, Erd\H{o}s extended the classical Hardy--Ramanujan formula for $p(n)$ by showing that $A$ has density $\alpha$ if and only if $\log p_A(n) \sim \log p(\alpha n)$. Nathanson asked if Erd\H{o}s'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 $$ \liminf_{n \rightarrow \infty} \frac{\log p_A(n)}{\log p(\alpha n)} \geq \left(\frac{\sqrt{6}}{\pi}-o_{\alpha}(1)\right)\log(1/\alpha)\;. $$ 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.