Density of large holes among power-free lattice points

Abstract

For fixed integers $d,r\ge 1$ with $dr\ge 2$, 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 $\exp \left\{-\frac{v_{d,q}{R}^d}{\zeta (dr)}\left[d(dr-1)\log R+dr\log \log R-{{\rm Y}}_{d,r,q}+r\frac{\log \log R}{\log R}-\frac{{\Lambda }_{d,r,q}}{\log R}+O_{d,r}\left(\frac{(\log \log R{)}^2}{(\log R{)}^2}\right)\right]\right\}$ as $R\to \infty$, where $v_{d,q}$ is the volume of the unit ${\ell }_q$-ball and ${{\rm Y}}_{d,r,q},{\Lambda }_{d,r,q}$ 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 $\exp \left\{-\frac{g}{\zeta (r)}\left[(r-1)\log g+r\log \log g-{\widehat{{\rm Y}}}_r+r\frac{\log \log g}{\log g}-\frac{{\hat{\Lambda }}_r}{\log g}+O_r\left(\frac{(\log \log g{)}^2}{(\log g{)}^2}\right)\right]\right\}$ as $g\to \infty$, where ${\widehat{{\rm Y}}}_r={{\rm Y}}_{1,r,1}+(r-1)\log 2$ and ${\hat{\Lambda }}_r={\Lambda }_{1,r,1}+r\log 2$. 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 $d=O_r((\log R{)}^r)$.

arXiv comment54 pages (including 4 appendices), 3 figures, 3 tables, ancillary files present