---
title: Uncovered Set in Covering Processes
url: https://www.emergentmind.com/topics/uncovered-set
type: topic
---

# Uncovered Set in Covering Processes

The **uncovered set** is the complement of the visited or covered region generated by a covering process. In the sources considered here, the term appears in several non-equivalent but structurally related settings: the set of vertices of \(\mathbb Z_n^d\) not visited by a random walk by time \(\alpha t_{\rm cov}\), the residual subset of the circle left outside random arcs in Dvoretzky-type coverings, the part of a bounded open set not covered by random Euclidean balls, and the set of integers omitted by a finite family of arithmetic progressions. The principal invariants attached to these complements are model-dependent and include total-variation distance, Hausdorff and Fourier dimensions, quantitative coverage probabilities, and asymptotic density [1911.05581], [2511.13068], [2112.14979], [1811.03547], [2511.09726].

## 1. Model-dependent definitions

The uncovered set is always defined as a complement, but the ambient space and the covering mechanism vary substantially. In the discrete torus setting, it is the set of sites not yet hit by a random walk. In Dvoretzky covering on the circle, it is the complement of a random arc covering, either in the limsup sense or as the set of points never hit by any arc. In Euclidean random covering, it is the residual region left outside a union of balls centered at i.i.d. samples. In arithmetic covering systems, it is the set of integers missed by a finite union of congruence classes.

| Setting | Ambient space | Uncovered set |
|---|---|---|
| Random walk | \(\mathbb Z_n^d\) | \(\mathcal U(t)=\{x:\tau_x>t\}\) |
| Dvoretzky covering | \(\mathbb T=\mathbb R/\mathbb Z\) | \(K_\ell=\mathbb T\setminus E_\ell\) or \(E=\mathbb T\setminus\bigcup_{n=1}^\infty I_n\) |
| Random geometric cover | \(E\subset\mathbb R^n\) | \(U=E\setminus\bigcup_{i=1}^N B(X_i,\delta)\) |
| Covering systems | \(\mathbb Z\) | \(U=\mathbb Z\setminus\bigcup_{i=1}^k A_i\) |

This variation in definition is reflected in the analytical questions asked about the complement. In some settings one studies the full law of the uncovered configuration; in others one studies dimension, multiplicity, or density. The common feature is that the complement records the residual structure after a stochastic or combinatorial covering mechanism has acted [1911.05581], [2511.13068], [2112.14979], [1811.03547], [2511.09726].

## 2. Uncovered vertices for random walk on \(\mathbb Z_n^d\)

Let \(X=(X(t))_{t\ge0}\) be a simple random walk on \(\mathbb Z_n^d\) with \(d\ge3\), started in stationarity. For each \(x\in\mathbb Z_n^d\), the hitting time is
\[
\tau_x=\inf\{\,t\ge0:\;X(t)=x\},
\]
and the expected cover time is
\[
t_{\rm cov}=\max_{x\in\mathbb Z_n^d}\E_x\bigl[\max_{y\in\mathbb Z_n^d}\tau_y\bigr].
\]
The uncovered set at time \(t\) is
\[
\mathcal U(t)=\{\,x\in\mathbb Z_n^d:\tau_x>t\},
\]
and for \(\alpha\in(0,1)\),
\[
\mathcal U_\alpha=\mathcal U(\alpha t_{\rm cov}).
\]

The central theorem states that for fixed \(d\ge3\), if \(\alpha_1(d)\) satisfies
\[
\alpha_1(d)>\frac{3}{4}\,\frac{\,d-\tfrac23\,}{\,d-1\,},
\]
then for each \(\alpha>\alpha_1(d)\), with
\[
p_{\alpha,n}=n^{-\alpha d}(1+o(1)),
\]
the law of \(\mathcal U_\alpha\) is asymptotically indistinguishable in total variation from an i.i.d. Bernoulli field on \(\mathbb Z_n^d\) with success probability \(p_{\alpha,n}\):
\[
d_{TV}\!\bigl(\Law(\mathcal U_\alpha),\,\nu_{\alpha,n}\bigr)
=\max_{A\subset\mathbb Z_n^d}\bigl|\P(\mathcal U_\alpha=A)-\nu_{\alpha,n}(A)\bigr|
\longrightarrow 0.
\]
The proof uses the Chen--Stein bound
\[
d_{TV}\bigl(\mathcal U_\alpha,\text{Bernoulli}(p_{\alpha,n})\bigr)\le 8\,(b_1+b_2+b_3),
\]
with
\[
b_1\asymp n^d\cdot n^{\gamma d}\,n^{-2\alpha d},
\]
\[
b_2\asymp n^d\bigl[n^{\zeta d}\,n^{-2\alpha d/(1+p_d)}+\;n^{\gamma d}\,n^{-2\alpha d}\bigr],
\]
\[
b_3\asymp n^d\,n^{-\alpha d}\,n^{-\gamma(d-2)/2}(\log n)^2,
\]
where \(p_d\) is the return probability in \(\mathbb Z^d\), \(\gamma=2\alpha-1-\varepsilon\), and \(\zeta\) is chosen small. Under the condition
\[
\alpha>\frac34\,\frac{\,d-\tfrac23\,}{\,d-1\,},
\]
one checks \(b_i=o(1)\), so the total-variation distance vanishes.

The proof strategy replaces the true uncovered set by an “after \(A\) excursions” proxy, chooses for each \(x\) a dependency neighborhood \(B_x\) of radius about \(100R\) with \(R=n^\gamma\), and controls \(b_1\), \(b_2\), and \(b_3\) using two-point estimates, mixing-time estimates, and a concentration inequality of Lezaud for the stationary chain of excursion-exit points. The key input for \(b_3\) is exponential concentration of
\[
\frac1A\sum_{i=1}^A\log f(Y_{i-1},Y_i)
\]
around its mean, where
\[
f(x,y)=\P\bigl(\text{walk does not hit center in one excursion }\bigm|X_{\rm in}=x,\;X_{\rm out}=y\bigr).
\]

Relative to the original Miller--Sousi argument, which produced a constant \(\alpha_1(d)\to1\) as \(d\to\infty\), the Chen--Stein argument yields
\[
\alpha_1(d)=\frac34\,\frac{\,d-\tfrac23\,}{\,d-1\,},
\qquad
\lim_{d\to\infty}\alpha_1(d)=\tfrac34.
\]
At times \(\alpha t_{\rm cov}\) with \(\alpha>3/4\), the set of unvisited vertices is, in total variation, indistinguishable from an i.i.d. sprinkling of intensity \(n^{-\alpha d}\); in particular, the uncovered points are uniformly scattered and exhibit no residual clustering or large-scale structure [1911.05581].

## 3. Dvoretzky random coverings on the circle

In the classical Dvoretzky model, one works on
\[
\mathbb T=\mathbb R/\mathbb Z\cong[0,1).
\]
Let \(\ell=(\ell_k)_{k\ge1}\) be a nonincreasing positive sequence with \(0<\ell_k<1\) and \(\sum_{k=1}^\infty \ell_k=\infty\), and let \(\omega_1,\omega_2,\dots\) be i.i.d. uniform points in \(\mathbb T\). The random arcs are
\[
I_k=(\omega_k,\;\omega_k+\ell_k)\subset\mathbb T,
\]
and the covering set is
\[
E_\ell=\limsup_{k\to\infty}I_k=\{\,t\in\mathbb T:\;t\in I_k\text{ i.o.}\}.
\]
Shepp’s criterion gives
\[
P(E_\ell=\mathbb T)=1
\quad\Leftrightarrow\quad
\sum_{k=1}^\infty k^{-2}\exp\bigl(\ell_1+\cdots+\ell_k\bigr)=\infty.
\]
In the complementary regime, the uncovered set
\[
K_\ell=\mathbb T\setminus E_\ell
\]
is almost surely nonempty and of Lebesgue measure zero. With
\[
D_\ell=\limsup_{k\to\infty}\frac{\ell_1+\cdots+\ell_k}{\log k},
\]
Kahane’s theorem yields
\[
\dim_{\mathcal H}K_\ell=1-D_\ell,
\]
and when \(\ell_k\sim\alpha/k\), one has \(D_\ell=\alpha\) and \(\dim_{\mathcal H}K_\ell=1-\alpha\). Theorem 1.2 strengthens this to the Salem property:
\[
\dim_{\mathcal F}(K_\ell)=\dim_{\mathcal H}(K_\ell)=1-D_\ell.
\]
Equivalently, one constructs a random probability measure \(\mu_{\mathrm{RC}}\) supported on \(K_\ell\) with sharp Fourier decay: for each \(\tau<1-D_\ell\),
\[
\bigl|\widehat\mu_{\mathrm{RC}}(n)\bigr|^2=O(n^{-\tau}),
\qquad n\to\infty.
\]
The construction is via
\[
X_k(t)=\frac{1-1_{I_k}(t)}{1-\ell_k},\qquad
M_k(t)=\prod_{j=1}^k X_j(t),\qquad
d\mu_k(t)=M_k(t)\,dt,
\]
where \((\mu_k)\) is a positive measure-valued martingale converging almost surely weakly to \(\mu_{\mathrm{RC}}\). The Fourier-dimension lower bound is obtained through a vector-valued martingale argument, Pisier’s martingale-type inequality in \(\ell^q\), a translation-cancellation trick for high frequencies, and \(L^1\)-modulus-of-continuity estimates for martingale differences [2511.13068].

A distinct but related formulation considers the non-covered set
\[
E=\mathbb T\setminus\bigcup_{n=1}^\infty I_n,
\]
that is, the set of points never hit by any arc. Under the assumptions
\[
(A)\quad\sum_{n=1}^\infty \ell_n=\infty,
\qquad
(B)\quad \sum_{n=1}^\infty(\ell_n-\ell_{n+1})\exp\!\bigl(\sum_{k=1}^n\ell_k\bigr)<\infty,
\]
this set is almost surely nonempty but has Lebesgue measure zero. The natural multiplicative chaos measure is defined by
\[
P_n(t)=\frac{1-1_{(\omega_n,\omega_n+\ell_n)}(t)}{1-\ell_n},
\qquad
M_n(t)=\prod_{k=1}^n P_k(t),
\qquad
\mu_n(dt)=M_n(t)\,dt,
\]
and converges to a nonzero random measure \(\mu_D\) supported on \(E\). Introducing the kernel
\[
K(t)=\exp\Bigl(\sum_{n=1}^\infty (\ell_n-\|t\|)_+\Bigr),
\]
one obtains: if
\[
\int_{\mathbb T}K(t)\,dt<\infty,
\]
then \(\mu_D*\mu_D\) is almost surely absolutely continuous, so by Riemann--Lebesgue,
\[
\widehat\mu_D(k)\to0
\qquad (|k|\to\infty).
\]
Thus \(\mu_D\) is a Rajchman measure and \(E\) is an \(M_0\)-set, hence a set of multiplicity. In the power-law case \(\ell_n=\alpha/n\), one convolution suffices when \(\alpha<1/2\); for \(\alpha\ge1/2\), higher convolutions may be taken so long as the \(d\)-fold kernel remains integrable [2511.09726].

## 4. Random geometric covers in bounded subsets of \(\mathbb R^n\)

Let \(E\subset\mathbb R^n\) be bounded and open, and let \(X_1,\dots,X_N\) be i.i.d. samples drawn uniformly from \(E\). The uncovered region is
\[
U=E\setminus\bigcup_{i=1}^N B(X_i,\delta).
\]
The geometric framework is built on positive reach and good partitions. For a closed set \(A\subset\mathbb R^n\), one defines
\[
\delta_A(x)=\operatorname{dist}(x,A),
\qquad
\operatorname{Unp}(A)=\{x\in\mathbb R^n:\text{ there is a unique }a\in A\text{ with }|x-a|=\delta_A(x)\},
\]
\[
\operatorname{reach}(A,a)=\sup\{r>0:B(a,r)\subset \operatorname{Unp}(A)\},
\qquad
\operatorname{reach}(A)=\inf_{a\in A}\operatorname{reach}(A,a).
\]
A finite measurable partition
\[
R_E=\{R_1,\dots,R_M\}
\]
is a good \((\varepsilon,\delta)\)-partition if \(\operatorname{diam}(R_j)\le\delta\) and \(|R_j|\ge\varepsilon\) for all \(j\).

If \(E^c\) has positive reach \(\rho>0\), then for every \(\delta\in(0,\rho)\) there is a finite partition \(R_E\) of \(E\) with
\[
\operatorname{diam}(R)\le3\delta,
\qquad
|R|\ge\delta^n/n^{n/2},
\]
and therefore
\[
P\bigl(E\subset\bigcup_{i=1}^N B(X_i,3\delta)\bigr)
\ge
1-M\exp\!\Bigl(-\,\frac{\delta^n}{n^{n/2}|E|}\,N\Bigr).
\]
If \(E=U\setminus A\) where \(\operatorname{reach}(U^c)>\rho>0\) and \(|A|<\delta^n/n^{n/2}\), then one similarly gets
\[
\inf_{R\in R_E}|R|\ge \delta^n/n^{n/2}-|A|>0,
\]
and
\[
P(E\subset\bigcup B(X_i,3\delta))
\ge
1-M\exp\!\Bigl(-\,\frac{\delta^n/n^{n/2}-|A|}{|E|}\,N\Bigr).
\]

The proof proceeds by constructing a Whitney-type decomposition of the \(\delta\)-interior
\[
E^*=\{x:\operatorname{dist}(x,\partial E)\ge\delta\}
\]
using dyadic cubes of side \(\ell=\delta/\sqrt n\). Cubes intersecting \(E^*\) lie entirely inside \(E\), have diameter at most \(\delta\), and volume \(\delta^n/n^{n/2}\). Fattening by \(B(0,\delta)\) yields overlapping patches of diameter at most \(3\delta\), from which a measurable partition is carved out. The coverage estimate follows from the union bound and the observation that a ball centered in a cell covers the entire cell.

In two dimensions, when \(E^c\) does not have positive reach, the multiscale flat norm provides a surrogate. If \(E\subset\mathbb R^2\) has finite perimeter, \(\lambda>\Lambda_E\), and \(\delta\) satisfies
\[
|S_\lambda|<\delta^2/2,
\qquad
\delta<1/(5\lambda),
\]
where \(S_\lambda\) is a minimizer of \(F_\lambda(\partial E)\), then the denoised set \(E_\lambda=E-S_\lambda\) has reach \(>1/(5\lambda)\) and admits a partition of cells of diameter \(\le3\delta\) and measure \(\ge\delta^2/2\). Intersecting those cells with \(E\) yields an \(\alpha\)-almost-partition with \(\alpha=\delta^2/(2|E|)\), and hence
\[
P\Bigl(|B(X,3\delta)\cap E|\ge(1-\alpha)|E|\Bigr)
\ge
1-M\exp\!\Bigl(-\,\frac{\delta^2/2-|S_\lambda|}{|A|}\,N\Bigr).
\]
These results make explicit how coverage depends on \(N\), \(\delta\), and the geometry of \(E\), and they provide exponentially small tail bounds for failure of complete or almost-complete coverage [2112.14979].

## 5. Uncovered density in arithmetic covering systems

A covering system is a finite family of arithmetic progressions
\[
\{A_i=a_i\pmod{d_i}\}_{i=1}^k
\]
whose union is all of \(\mathbb Z\). The uncovered set is
\[
U=\mathbb Z\setminus\bigcup_{i=1}^k\{\,n:\,n\equiv a_i\pmod{d_i}\},
\]
and its asymptotic density is
\[
\rho(U)=\liminf_{N\to\infty}\frac{|U\cap[-N,N]|}{2N+1}.
\]

The main theorem gives a sharp sufficient condition ensuring that \(\rho(U)\) stays bounded away from \(0\). The paper introduces a mildly growing multiplicative weight \(u\) on the moduli and sets
\[
C=\sum_{i=1}^k \frac{u(d_i)}{d_i}.
\]
For any \(\varepsilon>0\), there is \(M=M(\varepsilon)\) such that if the moduli are distinct, satisfy \(d_i>M\), and obey \(C<\infty\), then
\[
\rho(U)\ge \tfrac12\exp(-4C).
\]
When the moduli lie in \([n,Cn]\) with \(n\gg1\), this yields
\[
\rho(U)\ge \tfrac12\,C^{-4+o(1)}>0,
\]
confirming the Erdős--Graham conjecture for moduli in \([n,Cn]\).

The proof is a sieve in stages. Writing \(Q=\operatorname{lcm}(d_1,\dots,d_k)\) and listing its prime divisors \(p_1,\dots,p_n\), one exposes the congruence classes prime by prime. If \(R_i\) is the set of integers not yet covered after stage \(i\), and \(\alpha_i(x)\) is the fraction of the \(p_i\)-fiber at \(x\) that is removed at stage \(i\), then the first--second moment lemma bounds the removed mass by
\[
P_i(B_i)\le \min\Bigl\{\, \mathbb E_{i-1}\bigl[\alpha_i(x)\bigr]\,,\; \frac{\mathbb E_{i-1}\bigl[\alpha_i(x)^2\bigr]}{4\,d_i(1-d_i)} \Bigr\}.
\]
A change-of-measure argument then yields an explicit lower bound on \(\rho(U)\), from which the estimate \(\rho(U)\ge \tfrac12\exp(-4C)\) follows.

The method has several corollaries. It proves Schinzel’s conjecture that in any covering system there exists a pair of moduli, one of which divides the other. It improves Hough’s minimum-modulus bound from \(10^{16}\) to
\[
\min_i d_i\le 6.16\times10^5.
\]
It also shows that no covering with all moduli odd and square-free can exist, and reproves that in any covering one modulus must be divisible by \(2\) or \(3\) [1811.03547].

## 6. Analytical themes and significance

Although these uncovered sets live in different spaces and are measured by different invariants, the proofs share a consistent structural pattern: one isolates the residual complement, quantifies the dependence induced by the covering mechanism, and then passes from local control to a global description. For random walk on \(\mathbb Z_n^d\), this takes the form of Chen--Stein approximation with local dependency neighborhoods and a spectral concentration estimate. For Dvoretzky coverings, it takes the form of multiplicative chaos martingales, Fourier decay, and translation-cancellation arguments. For geometric random covers, it is a Whitney-type partition plus a union bound. For arithmetic covering systems, it is a prime-by-prime sieve with first--second moment control [1911.05581], [2511.13068], [2112.14979], [1811.03547].

The behavior of the uncovered set also varies sharply with the model. In high-dimensional random walk, the uncovered configuration becomes asymptotically Bernoulli and uniformly scattered. In Dvoretzky covering, the uncovered set is Lebesgue-null but can still be a Salem set or support a Rajchman measure, so its harmonic-analytic size matches or complements its geometric thinness. In Euclidean random covering, the relevant issue is quantitative near-complete coverage under geometric regularity assumptions such as positive reach. In covering systems, the central object is not dimension but residual density, and the uncovered set becomes the obstruction to exact coverage.

Taken together, these results show that the uncovered set is a central residual object in covering theory rather than a secondary remainder. Depending on the ambient model, it encodes late points of a walk, fractal leftovers of random arcs, geometric holes in random sampling, or arithmetic failures of congruence coverings. The corresponding descriptors—total variation, Hausdorff and Fourier dimensions, coverage probability, and density—form the natural taxonomy of the subject [2511.09726].

Source: https://www.emergentmind.com/topics/uncovered-set