arXiv 2104.026922021-04-06math.NTmath.CO

限制分拆的精确极限定理

Exact Limit Theorems for Restricted Integer Partitions

Asaf Cohen Antonir, Asaf Shapira

摘要：

对于一组正整数$A$，令$p_A(n)$表示将$n$写成来自$A$的整数之和的方式的数量，并且令$p(n)$表示通常的分区函数。在40年代早期，Erdens扩展了经典的Hardy-Ramanujan公式，证明了$A$有密度$α$当且仅当$\log p_A(n) \sim \log p(αn)$。Nathanson问Erdans定理是否也适用于$A$的低密度，即$A$是否有低密度$α$当且仅当$\log p_A(n) / \log p(αn)$有下限$1$。我们回答这个问题否定的建设，为每一个$α> 0$，一组整数$A$的密度较低$α$，满足$$ \liminf_{n \rightarrow \infty} \frac{\log p_A(n)}{\log p(αn)} \geq \left(\frac{\sqrt{6}}π-o_α(1)\right)\log(1/α)\;. $$我们进一步表明，上述界是最好的可能（到$o_α(1)$长期），从而确定确切的极值关系之间的一组整数的密度较低，其分区函数的下限。我们还证明了一个类似的定理就上密度的一组整数，回答另一个问题的Nathanson。

英文摘要：

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, Erdős extended the classical Hardy--Ramanujan formula for $p(n)$ by showing that $A$ has density $α$ if and only if $\log p_A(n) \sim \log p(αn)$. 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 $\log p_A(n) / \log p(αn)$ 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 $$ \liminf_{n \rightarrow \infty} \frac{\log p_A(n)}{\log p(αn)} \geq \left(\frac{\sqrt{6}}π-o_α(1)\right)\log(1/α)\;. $$ We further show that the above bound is best possible (up to the $o_α(1)$ 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.

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