Density of large holes among power-free lattice points

Abstract

For fixed integers $d,r\ge1$ with $dr\ge2$, the $r$-free points of $\mathbb{Z}^d$ are the vectors whose coordinate gcd is not divisible by the $r$th power of any prime. We consider the set $W_{d,r}$ of non-$r$-free points in $\mathbb{Z}^d$ as the vertex set of a graph with nearest-neighbour adjacency. For $1\le q\le\infty$, we say that a point $t\in\mathbb Z^d$ is $R$-deep if $t+\{n\in\mathbb Z^d:\|n\|_q\le R\}\subseteq W_{d,r}$. Thus $R$-deep points are the lattice centres of closed $\ell_q$-balls whose lattice points are all non-$r$-free. We call a connected component $R$-large if it contains an $R$-deep point. We mark each finite $R$-large component by selecting its lexicographically least $R$-deep point as its representative. Uniformly for $1\le q\le\infty$, we show that the density of $R$-deep points and the density of these representatives both equal $$ \exp\!\left\{ -\frac{v_{d,q}}{ζ(dr)} \left(d(dr-1)R^d\log R+drR^d\log\log R\right) +O_{d,r}(R^d) \right\} $$ as $R\to\infty$, where $v_{d,q}$ denotes the volume of the unit ball in $(\mathbb R^d,\|\cdot\|_q)$. 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. When $d=1$, we show that the density of $r$-free integers followed by exactly $g-1$ non-$r$-free integers and then another $r$-free integer is $$\exp\!\left\{-\frac{r-1}{ζ(r)}g\log g -\frac{r}{ζ(r)}g\log\log g+O_r(g)\right\}$$ as $g\to\infty$, improving the asymptotic $\exp\left\{-\left(\frac{r-1}{ζ(r)}+o(1)\right) g\log g\right\}$ which follows from Grimmett's work. % For fixed $r$, we also obtain estimates uniform in growing dimensions $d=O_r((\log R)^r)$.