对角有理分拆的算术约束与极限定律
Arithmetic Constraints and Limit Laws for Diagonal Rational Partitions
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中文总结 AI 辅助
本文研究对角有理分拆的计数渐近与极限分布，给出主项及修正项，并证明素数分母块总和的阈值行为及残差分布的高斯极限。
中文摘要 AI 辅助
设$R_N(m)$统计将$m$分拆为分子分母均不超过$N$的既约正分数（不含整数部分）的无序分拆数。一致地对紧致正区间内的$\rho$，有$\log R_N(\lfloor\rho N\rfloor)=\sqrt{2\rho}\\,N^{3/2}-\kappa(\rho)N^{3/2}/\log N+o(N^{3/2}/\log N)$。正且连续可微的函数$\kappa$是格距积分的显式求和，其中$\kappa(1)\approx 0.00264713$。对于$n$的一致分拆，在$(n/2,n]$中除$o_{\mathbb{P}}(n/\log n)$个素数分母块外，所有块的总和为2或3，具体取决于$p/n$是低于还是高于接近0.7522的显式阈值。对于固定的$N$，我们给出Ehrhart分子并确定其极点如何控制拟多项式系数。其残差分布在$N\ge 4$时不对称，但在低于$\lceil N/2\rceil$阶的所有矩上与独立模型一致。我们识别出首次偏差并证明高斯极限。我们还获得了两种抽样规则下的联合分母与大小定律。
英文摘要
Let $R_N(m)$ count unordered partitions of $m$ into reduced positive fractions whose numerators and denominators are at most $N$, excluding integer parts. Uniformly for $ρ$ in compact positive intervals, $\log R_N(\lfloorρN\rfloor)=\sqrt{2ρ}\,N^{3/2}-κ(ρ)N^{3/2}/\log N+o(N^{3/2}/\log N)$. The positive, continuously differentiable function $κ$ is an explicit sum of lattice-distance integrals, with $κ(1)\approx 0.00264713$. For a uniform partition of $n$, all but $o_{\mathbb{P}}(n/\log n)$ prime-denominator blocks in $(n/2,n]$ have total 2 or 3, according to whether $p/n$ lies below or above an explicit threshold near 0.7522. For fixed $N$, we give the Ehrhart numerator and determine how its poles control quasipolynomial coefficients. Its residue distribution is asymmetric for $N\ge 4$, but agrees with an independent model in every moment below order $\lceil N/2\rceil$. We identify the first discrepancy and prove a Gaussian limit. We also obtain joint denominator and size laws under two sampling rules.
发表机构
- Pie Mathematics Association（Pie数学协会）
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