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Density of large holes among power-free lattice points

This paper establishes precise asymptotic formulas for the densities of large holes and deep points within $r$ -free lattice points in $\mathbb{Z}^d$ , refining previous bounds and exact-gap expansions by providing explicit constants and error terms that hold uniformly across dimensions and ball norms.

Original authors: Francesco Cellarosi

Published 2026-10-02

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Francesco Cellarosi

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the vast, orderly grid of the integers, there exists a hidden pattern of exclusion. Imagine a vast field of points stretching infinitely in every direction, where each point is defined by a set of whole number coordinates. Some of these points are "visible" from the center, meaning their coordinates share no common divisor other than one. Others are "invisible" because their coordinates share a common factor, such as being all even or all multiples of three. Mathematicians call the visible ones "primitive" or "power-free" points, and the invisible ones "non-power-free." For decades, researchers have studied how these invisible points cluster together. They know that if you look at a small patch of the grid, you will almost always find at least one visible point. But what happens if you look at a very large patch? Is it possible to find a massive, empty region where every single point is invisible?

This question leads to the study of "holes" in the lattice. A hole is a large, connected area where no visible points exist. The size of such a hole is measured by how far you can travel from its center in any direction before hitting a visible point. For a long time, mathematicians knew that these holes could be made arbitrarily large, but they did not know how rare they were. They could prove that a hole of a certain size exists, but they could not calculate the precise probability of finding one in a random section of the grid. The question was not just about existence, but about density: as the size of the hole grows, how quickly does the likelihood of finding one drop to zero?

A new study by Francesco Cellarosi addresses this problem with remarkable precision. The research focuses on a specific type of invisible point: those whose coordinates are all divisible by the same prime number raised to a certain power. By treating the grid as a graph where points are connected to their immediate neighbors, the author identifies the "deep" points—the centers of these massive invisible regions. The study calculates the exact rate at which the density of these deep points decreases as the size of the hole increases. The result is a detailed formula that describes this drop-off, revealing that the probability of finding such a hole is not just small, but follows a very specific, predictable curve involving logarithms and powers.

The paper does more than just count these holes; it distinguishes between different ways of measuring their size. The author considers holes shaped like spheres, cubes, and diamonds, showing that while the shape changes the specific numbers in the formula, the fundamental behavior remains the same. The study also refines previous work by other mathematicians who had established that these holes exist and provided rough upper limits on their frequency. Cellarosi's work sharpens these limits, providing a much more accurate picture of the landscape. The findings show that the density of these large holes is governed by a complex interplay between the size of the grid, the power of the prime numbers involved, and the geometry of the region being examined.

One of the most significant aspects of this research is its application to one-dimensional number lines, which corresponds to the study of integers rather than multi-dimensional grids. In this simpler case, the problem translates to finding long sequences of consecutive integers that are all "non-power-free." For example, finding a run of numbers where none of them are "square-free" (meaning each is divisible by a square number like 4, 9, or 16). The study provides a precise formula for the density of such runs as they get longer. This improves upon earlier estimates by other researchers, offering a clearer understanding of how these gaps behave as they stretch out. The author proves that the probability of finding a gap of a specific length follows a specific mathematical pattern, correcting and refining earlier approximations that were less precise.

The methodology relies on a clever combination of number theory and probability. The author constructs a "random covering model," a theoretical framework where the invisible points are generated by a random process that mimics the actual distribution of numbers. By analyzing this random model, the author can calculate the probability of a large hole appearing. The proof involves carefully counting how many points in a large region can be covered by the "shadows" of small prime numbers and then determining how many large prime numbers are needed to cover the remaining points. This process allows the author to derive the exact asymptotic form of the density, which describes the behavior of the system as the hole size approaches infinity.

The results are uniform across different dimensions, meaning the same underlying logic applies whether the grid is a line, a plane, or a higher-dimensional space. The study also explores what happens when the dimension of the grid grows alongside the size of the hole, a scenario that was previously difficult to analyze. The author shows that even in these high-dimensional settings, the density of large holes follows a predictable pattern, though the specific constants change. This suggests a deep structural consistency in how these invisible clusters form, regardless of the complexity of the space they inhabit.

In the context of the broader mathematical landscape, this work connects to the study of "visible" lattice points, which are points that can be seen from the origin without any other lattice point blocking the view. The invisible points are the ones that block the view. The existence of large holes among the invisible points means there are vast regions where the view is completely blocked. The study confirms that while these regions are rare, they are not impossible, and their frequency can be calculated with high precision. The author's work builds on earlier constructions by other mathematicians who used the Chinese Remainder Theorem to prove that such holes must exist, but goes further by quantifying exactly how often they occur.

The paper also addresses the relationship between the density of these holes and the density of the points that define them. It shows that the probability of finding a hole is related to the volume of the region and the density of the invisible points, but with a correction factor that accounts for the specific geometry of the problem. This correction factor involves constants that depend on the dimension of the grid and the power of the prime numbers. The author provides explicit formulas for these constants, allowing for precise calculations in specific cases, such as the two-dimensional plane or the one-dimensional number line.

Ultimately, the study provides a comprehensive map of the "holes" in the lattice of power-free points. It transforms a qualitative understanding of these gaps into a quantitative science, offering a formula that predicts their frequency with high accuracy. The work does not just confirm that these holes exist; it tells us exactly how rare they are and how their rarity changes as they grow larger. This level of detail is a significant step forward in the field, providing a solid foundation for future research into the distribution of numbers and the geometry of lattices. The findings are rigorous and proven, offering a definitive answer to a question that had previously only been partially understood.

Technical Summary: Density of Large Holes Among Power-Free Lattice Points

Problem Statement
The paper investigates the statistical distribution of "large holes" within the set of $r$ -free lattice points in $\mathbb{Z}^d$ . An $r$ -free lattice point is defined as a vector whose coordinate greatest common divisor is not divisible by the $r$ -th power of any prime. The complement set, $W_{d,r}$ , consists of non- $r$ -free points. The central object of study is the density of finite connected components of $W_{d,r}$ that contain a "deep" interior.

Specifically, for a given radius $R$ and an $\ell_q$ -norm, a point $t \in \mathbb{Z}^d$ is called $R$ -deep if the closed $\ell_q$ -ball of radius $R$ centered at $t$ is entirely contained within $W_{d,r}$ . The paper analyzes the asymptotic behavior of the density of such $R$ -deep points ( $\eta_{d,r,q}(R)$ ), as well as the densities of representatives for finite components containing such points ( $\lambda_{d,r,q}(R)$ and $\kappa_{d,r,q}(R)$ ), as $R \to \infty$ . The study covers fixed dimensions $d$ and exponents $r$ (with $dr \ge 2$ ), as well as regimes where the dimension $d$ grows with $R$ .

Methodology
The author employs a combination of probabilistic number theory, analytic number theory, and geometric combinatorics. The core methodology involves:

Random Covering Model: The deterministic set of non- $r$ -free points is compared to a random model where, for each prime $p$ , a residue class modulo $p^r$ is chosen uniformly and independently. The probability of a specific finite pattern in this random model equals the density of that pattern in the deterministic set (via inclusion-exclusion and the Chinese Remainder Theorem).
Two-Stage Prime Analysis: The covering of a "window" (an $\ell_q$ $ℓ_{q}$ -ball of radius $R$ $R$ ) is analyzed by separating primes into "small" and "large" categories:
- Small Primes ( $p^r \le 2R$ ): The number of points in the window left uncovered by small primes is estimated using truncated inclusion-exclusion and bounds on the capacity of cosets. This establishes that the number of uncovered points is asymptotically $N/\zeta(dr)$ , where $N$ is the volume of the window.
- Large Primes ( $p^r > 2R$ ): The remaining uncovered points must be covered by distinct large primes. The probability of this event is estimated using sums over injective assignments of points to primes, denoted as $Z_{s,K}(L)$ . The asymptotic expansion of $-\log Z_{s,K}(L)$ is derived using techniques involving Bernoulli random variables, saddle-point methods, and precise estimates of prime sums.
Geometric Confinement: To ensure that a large hole corresponds to a finite connected component (rather than an infinite one), the author constructs a surrounding "shell" of $r$ -free points. This shell prevents paths from the hole to infinity, ensuring the component is finite and thus possesses a representative. The cost of enforcing this shell condition is shown to be negligible in the logarithmic density scale.
Error Control: The paper provides rigorous error terms involving powers of $\log R$ and $\log \log R$ , ensuring uniformity with respect to the norm parameter $q$ ( $1 \le q \le \infty$ ).

Key Contributions and Results

Asymptotic Formula for Hole Density: The primary result (Theorem 1.7) establishes that for fixed $d, r$ , the negative logarithm of the densities $\xi \in \{\lambda, \kappa, \eta\}$ admits the following asymptotic expansion as $R \to \infty$ :
$-\log \xi_{d,r,q}(R) = a_{d,r,q} R^d \left[ d(dr-1)\log R + dr \log \log R - \Upsilon_{d,r,q} + \frac{r \log \log R - \Lambda_{d,r,q}}{\log R} + O\left(\frac{(\log \log R)^2}{(\log R)^2}\right) \right]$
where $a_{d,r,q}$ is a volume factor, and $\Upsilon_{d,r,q}, \Lambda_{d,r,q}$ are explicit constants depending on $d, r, q$ . This refines previous upper bounds and positive-density constructions.
One-Dimensional Case (Gaps between $r$ -free integers): Specializing to $d=1$ , the paper derives the density of runs of non- $r$ -free integers. For a gap of length $g$ , the density of $r$ -free integers followed by exactly $g-1$ non- $r$ -free integers (and then another $r$ -free integer) is given by a sharp asymptotic formula (Theorem 1.11). This improves upon Grimmett's leading term and refines recent preprints by Jiang by providing explicit lower-order terms and error bounds.
High-Dimensional Regime: The paper extends the analysis to dimensions $d$ growing with $R$ (specifically $d = O((\log R)^r)$ ). It shows that the leading term of the negative logarithm of the density is $\Phi_{dr}(N/\zeta(dr))$ , where $\Phi_s(K) \approx (s-1)K \log K$ .
Comparison with Independent Models: In high dimensions, the paper compares the actual density of large holes to a model where lattice points are independently non- $r$ -free with probability $1 - \zeta(dr)^{-1}$ . It finds that the actual probability of a large hole is significantly smaller than the independent model predicts; specifically, the ratio of their negative logarithms grows as $\log N / \log 2$ . This highlights the strong correlations inherent in the arithmetic structure of power-free integers.

Significance and Claims
The paper claims to provide the first sharp asymptotic formulas for the density of large holes among power-free lattice points, including precise lower-order terms and error estimates.

It sharpens the positive-density hole constructions of Baake, Moody, and Pleasants, and Pleasants and Huck, by quantifying the decay rate of the density as the hole size increases.
It refines the understanding of gap statistics for $r$ -free integers, offering a more detailed expansion than previous probabilistic representations (Grimmett) and correcting/improving upon recent preprints (Jiang) regarding the specific form of the error terms and the distinction between "at least" and "exactly" gap densities.
It demonstrates that the arithmetic constraints of the sieve (power-free condition) create a "repulsion" effect that makes large holes much rarer than in a purely random (independent) setting, a phenomenon that becomes increasingly pronounced in higher dimensions.

The work is presented as a rigorous analytic number theory result, relying on established tools (Chinese Remainder Theorem, inclusion-exclusion, prime number theorem estimates) applied to a specific geometric sieve problem, without proposing new experimental applications or future directions beyond the mathematical scope of the derived asymptotics.

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