幂自由格点中大洞的密度
Density of large holes among power-free lattice points
- Queen’s University（女王大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文研究了整数格点中非幂自由点形成的连通分量，证明了深点及其代表点的密度渐近公式，并改进了相关上界和一元情形下的精确渐近。
AI中文摘要：
对于固定的整数$d,r\ge1$且$dr\ge2$，$\mathbb{Z}^d$中的$r$-自由点是那些坐标最大公约数不被任何素数的$r$次幂整除的向量。我们将$\mathbb{Z}^d$中非$r$-自由点的集合$W_{d,r}$视为一个具有最近邻邻接关系的图的顶点集。对于$1\le q\le\infty$，如果$t+\{n\in\mathbb Z^d:\\|n\\|_q\le R\}\subseteq W_{d,r}$，则称点$t\in\mathbb Z^d$是$R$-深的。因此，$R$-深点是闭$\ell_q$-球的格点中心，其所有格点都是非$r$-自由的。如果连通分量包含一个$R$-深点，我们称其为$R$-大的。我们通过选择其字典序最小的$R$-深点作为代表来标记每个有限的$R$-大分量。对于$1\le q\le\infty$一致地，我们证明当$R\to\infty$时，$R$-深点的密度和这些代表的密度都等于$$ \exp\\!\left\{ -\frac{v_{d,q}}{\zeta(dr)} \left(d(dr-1)R^d\log R+drR^d\log\log R\right) +O_{d,r}(R^d) \right\} $$，其中$v_{d,q}$表示$\mathbb R^d$中单位球在$\\|\cdot\\|_q$范数下的体积。对于深点，这改进了Baake、Moody和Pleasants以及Pleasants和Huck的正密度洞构造，以及后两位作者关于稀疏模式频率的上界。当$d=1$时，我们证明$r$-自由整数后面恰好跟着$g-1$个非$r$-自由整数然后再跟一个$r$-自由整数的密度为$$\exp\\!\left\{-\frac{r-1}{\zeta(r)}g\log g -\frac{r}{\zeta(r)}g\log\log g+O_r(g)\right\}$$当$g\to\infty$时，改进了从Grimmett的工作中得出的渐近式$\exp\left\{-\left(\frac{r-1}{\zeta(r)}+o(1)\right) g\log g\right\}$。对于固定的$r$，我们还获得了在增长维度$d=O_r((\log R)^r)$下一致成立的估计。
英文摘要：
For fixed integers $d,r\ge1$ with $dr\ge2$, the $r$-free points of $\mathbb{Z}^d$ are those whose coordinate gcd is not divisible by the $r$th power of any prime. Fix $1\le q\le \infty$. A lattice point is $R$-deep if every lattice point in the closed $\ell_q$-ball of radius $R$ centred there is non-$r$-free. We mark each finite nearest-neighbour component of non-$r$-free points containing an $R$-deep point by its lexicographically least such point. Uniformly for $1\le q\le\infty$, the densities of $R$-deep points and of these representatives both have the asymptotic form $\exp\left\{-\frac{v_{d,q}R^d}{ζ(dr)}\left[d(dr-1)\log R+dr\log\log R-Υ_{d,r,q}+r\frac{\log\log R}{\log R}-\frac{Λ_{d,r,q}}{\log R}+O_{d,r}\left(\frac{(\log\log R)^2}{(\log R)^2}\right)\right]\right\}$ as $R\to\infty$, where $v_{d,q}$ is the volume of the unit $\ell_q$-ball and $Υ_{d,r,q},Λ_{d,r,q}$ are explicit constants. For deep points, this sharpens the positive-density hole constructions of Baake, Moody, and Pleasants and of Pleasants and Huck, and the latter authors' upper bounds for sparse-pattern frequencies. For $d=1$, consider the densities of $r$-free integers followed by at least $g-1$ consecutive non-$r$-free integers, or by exactly $g-1$ such integers and then another $r$-free integer. Both have the asymptotics $\exp\left\{-\frac{g}{ζ(r)}\left[(r-1)\log g+r\log\log g-\widehatΥ_r+r\frac{\log\log g}{\log g}-\frac{\widehatΛ_r}{\log g}+O_r\left(\frac{(\log\log g)^2}{(\log g)^2}\right)\right]\right\}$ as $g\to\infty$, where $\widehatΥ_r=Υ_{1,r,1}+(r-1)\log2$ and $\widehatΛ_r=Λ_{1,r,1}+r\log2$. The exact-gap expansion improves Grimmett's leading asymptotic and refines the formula in Jiang's recent preprint. For fixed $r$, we also obtain estimates uniform in growing dimensions $d=O_r((\log R)^r)$.
