Density of large holes among power-free lattice points
Abstract
For fixed integers d,r≥1 with dr≥2, the r-free points of ℤd are those whose coordinate gcd is not divisible by the rth power of any prime. Fix 1≤ q≤ ∞. A lattice point is R-deep if every lattice point in the closed ℓq-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≤ q≤∞, the densities of R-deep points and of these representatives both have the asymptotic form exp{−vd,qRd/ζ(dr)[d(dr−1)log R+drloglog R−Υd,r,q+r(loglog R)/(log R)−Λd,r,q/(log R)+Od,r(((loglog R)2)/((log R)2))]} as R→∞, where vd,q is the volume of the unit ℓq-ball and Υd,r,q,Λd,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{−g/ζ(r)[(r−1)log g+rloglog g−Υ̂r+r(loglog g)/(log g)−Λ̂r/(log g)+Or(((loglog g)2)/((log g)2))]} as g→∞, where Υ̂r=Υ1,r,1+(r−1)log2 and Λ̂r=Λ1,r,1+rlog2. 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=Or((log R)r).
Related
- Density and holes of the visible points in certain cut-and-project sets
- Lattice points in high-dimensional ℓq balls with small radii
- Lattice points close to a smooth curve and squarefull numbers in short intervals. II
- The variance of the number of lattice points in narrow sectors
- Potential density of rational points on sextic double solids
BibTeX
@misc{cellarosi2026density,
title = {{Density of large holes among power-free lattice points}},
author = {Francesco Cellarosi},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2609.25457},
eprint = {2609.25457},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25457},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Density of large holes among power-free lattice points
AU - Francesco Cellarosi
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2609.25457
UR - https://arxiv.org/abs/2609.25457
AB - For fixed integers d,r≥1 with dr≥2, the r-free points of ℤ^d are those whose coordinate gcd is not divisible by the rth power of any prime. Fix 1≤ q≤ ∞. A lattice point is R-deep if every lattice point in the closed ℓ_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≤ q≤∞, the densities of R-deep points and of these representatives both have the asymptotic form exp{−v_d,qR^d/ζ(dr)[d(dr−1)log R+drloglog R−Υ_d,r,q+r(loglog R)/(log R)−Λ_d,r,q/(log R)+O_d,r(((loglog R)^2)/((log R)^2))]} as R→∞, where v_d,q is the volume of the unit ℓ_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{−g/ζ(r)[(r−1)log g+rloglog g−Υ̂_r+r(loglog g)/(log g)−Λ̂_r/(log g)+O_r(((loglog g)^2)/((log g)^2))]} as g→∞, where Υ̂_r=Υ_1,r,1+(r−1)log2 and Λ̂_r=Λ_1,r,1+rlog2. 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).
ER -