Uncovered Set in Covering Processes

• Uncovered set is the complement of the region covered by a stochastic or combinatorial process, defined differently in settings such as random walks, Dvoretzky coverings, geometric sampling, and arithmetic systems.
• The analysis employs methods like the Chen–Stein bound, martingale constructions, and sieve techniques to quantify properties such as total variation, Hausdorff/Fourier dimensions, and asymptotic density.
• These studies reveal that uncovered sets capture key residual structures, exhibiting uniform scattering in random walks, fractal properties in circle coverings, and density obstructions in arithmetic systems.

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 (Olesker-Taylor et al., 2019, Chen et al., 17 Nov 2025, Alvarado et al., 2021, Balister et al., 2018, Tan, 12 Nov 2025).

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	$\mathbb Z$	$\alpha t_{\rm cov}$0

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 (Olesker-Taylor et al., 2019, Chen et al., 17 Nov 2025, Alvarado et al., 2021, Balister et al., 2018, Tan, 12 Nov 2025).

2. Uncovered vertices for random walk on $\alpha t_{\rm cov}$1

Let $\alpha t_{\rm cov}$2 be a simple random walk on $\alpha t_{\rm cov}$3 with $\alpha t_{\rm cov}$4, started in stationarity. For each $\alpha t_{\rm cov}$5, the hitting time is

$\alpha t_{\rm cov}$6

and the expected cover time is

$\alpha t_{\rm cov}$7

The uncovered set at time $\alpha t_{\rm cov}$8 is

$\alpha t_{\rm cov}$9

and for $\mathbb Z_n^d$0,

$\mathbb Z_n^d$1

The central theorem states that for fixed $\mathbb Z_n^d$2, if $\mathbb Z_n^d$3 satisfies

$\mathbb Z_n^d$4

then for each $\mathbb Z_n^d$5, with

$\mathbb Z_n^d$6

the law of $\mathbb Z_n^d$7 is asymptotically indistinguishable in total variation from an i.i.d. Bernoulli field on $\mathbb Z_n^d$8 with success probability $\mathbb Z_n^d$9: $\mathcal U(t)=\{x:\tau_x>t\}$0 The proof uses the Chen--Stein bound

$\mathcal U(t)=\{x:\tau_x>t\}$1

with

$\mathcal U(t)=\{x:\tau_x>t\}$2

$\mathcal U(t)=\{x:\tau_x>t\}$3

$\mathcal U(t)=\{x:\tau_x>t\}$4

where $\mathcal U(t)=\{x:\tau_x>t\}$5 is the return probability in $\mathcal U(t)=\{x:\tau_x>t\}$6, $\mathcal U(t)=\{x:\tau_x>t\}$7, and $\mathcal U(t)=\{x:\tau_x>t\}$8 is chosen small. Under the condition

$\mathcal U(t)=\{x:\tau_x>t\}$9

one checks $\mathbb T=\mathbb R/\mathbb Z$0, so the total-variation distance vanishes.

The proof strategy replaces the true uncovered set by an “after $\mathbb T=\mathbb R/\mathbb Z$1 excursions” proxy, chooses for each $\mathbb T=\mathbb R/\mathbb Z$2 a dependency neighborhood $\mathbb T=\mathbb R/\mathbb Z$3 of radius about $\mathbb T=\mathbb R/\mathbb Z$4 with $\mathbb T=\mathbb R/\mathbb Z$5, and controls $\mathbb T=\mathbb R/\mathbb Z$6, $\mathbb T=\mathbb R/\mathbb Z$7, and $\mathbb T=\mathbb R/\mathbb Z$8 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 $\mathbb T=\mathbb R/\mathbb Z$9 is exponential concentration of

$K_\ell=\mathbb T\setminus E_\ell$0

around its mean, where

$K_\ell=\mathbb T\setminus E_\ell$1

Relative to the original Miller--Sousi argument, which produced a constant $K_\ell=\mathbb T\setminus E_\ell$2 as $K_\ell=\mathbb T\setminus E_\ell$3, the Chen--Stein argument yields

$K_\ell=\mathbb T\setminus E_\ell$4

At times $K_\ell=\mathbb T\setminus E_\ell$5 with $K_\ell=\mathbb T\setminus E_\ell$6, the set of unvisited vertices is, in total variation, indistinguishable from an i.i.d. sprinkling of intensity $K_\ell=\mathbb T\setminus E_\ell$7; in particular, the uncovered points are uniformly scattered and exhibit no residual clustering or large-scale structure (Olesker-Taylor et al., 2019).

3. Dvoretzky random coverings on the circle

In the classical Dvoretzky model, one works on

$K_\ell=\mathbb T\setminus E_\ell$8

Let $K_\ell=\mathbb T\setminus E_\ell$9 be a nonincreasing positive sequence with $E=\mathbb T\setminus\bigcup_{n=1}^\infty I_n$0 and $E=\mathbb T\setminus\bigcup_{n=1}^\infty I_n$1, and let $E=\mathbb T\setminus\bigcup_{n=1}^\infty I_n$2 be i.i.d. uniform points in $E=\mathbb T\setminus\bigcup_{n=1}^\infty I_n$3. The random arcs are

$E=\mathbb T\setminus\bigcup_{n=1}^\infty I_n$4

and the covering set is

$E=\mathbb T\setminus\bigcup_{n=1}^\infty I_n$5

Shepp’s criterion gives

$E=\mathbb T\setminus\bigcup_{n=1}^\infty I_n$6

In the complementary regime, the uncovered set

$E=\mathbb T\setminus\bigcup_{n=1}^\infty I_n$7

is almost surely nonempty and of Lebesgue measure zero. With

$E=\mathbb T\setminus\bigcup_{n=1}^\infty I_n$8

Kahane’s theorem yields

$E=\mathbb T\setminus\bigcup_{n=1}^\infty I_n$9

and when $E\subset\mathbb R^n$0, one has $E\subset\mathbb R^n$1 and $E\subset\mathbb R^n$2. Theorem 1.2 strengthens this to the Salem property: $E\subset\mathbb R^n$3 Equivalently, one constructs a random probability measure $E\subset\mathbb R^n$4 supported on $E\subset\mathbb R^n$5 with sharp Fourier decay: for each $E\subset\mathbb R^n$6,

$E\subset\mathbb R^n$7

The construction is via

$E\subset\mathbb R^n$8

where $E\subset\mathbb R^n$9 is a positive measure-valued martingale converging almost surely weakly to $U=E\setminus\bigcup_{i=1}^N B(X_i,\delta)$0. The Fourier-dimension lower bound is obtained through a vector-valued martingale argument, Pisier’s martingale-type inequality in $U=E\setminus\bigcup_{i=1}^N B(X_i,\delta)$1, a translation-cancellation trick for high frequencies, and $U=E\setminus\bigcup_{i=1}^N B(X_i,\delta)$2-modulus-of-continuity estimates for martingale differences (Chen et al., 17 Nov 2025).

A distinct but related formulation considers the non-covered set

$U=E\setminus\bigcup_{i=1}^N B(X_i,\delta)$3

that is, the set of points never hit by any arc. Under the assumptions

$U=E\setminus\bigcup_{i=1}^N B(X_i,\delta)$4

this set is almost surely nonempty but has Lebesgue measure zero. The natural multiplicative chaos measure is defined by

$U=E\setminus\bigcup_{i=1}^N B(X_i,\delta)$5

and converges to a nonzero random measure $U=E\setminus\bigcup_{i=1}^N B(X_i,\delta)$6 supported on $U=E\setminus\bigcup_{i=1}^N B(X_i,\delta)$7. Introducing the kernel

$U=E\setminus\bigcup_{i=1}^N B(X_i,\delta)$8

one obtains: if

$U=E\setminus\bigcup_{i=1}^N B(X_i,\delta)$9

then $\mathbb Z$0 is almost surely absolutely continuous, so by Riemann--Lebesgue,

$\mathbb Z$1

Thus $\mathbb Z$2 is a Rajchman measure and $\mathbb Z$3 is an $\mathbb Z$4-set, hence a set of multiplicity. In the power-law case $\mathbb Z$5, one convolution suffices when $\mathbb Z$6; for $\mathbb Z$7, higher convolutions may be taken so long as the $\mathbb Z$8-fold kernel remains integrable (Tan, 12 Nov 2025).

4. Random geometric covers in bounded subsets of $\mathbb Z$9

Let $\alpha t_{\rm cov}$00 be bounded and open, and let $\alpha t_{\rm cov}$01 be i.i.d. samples drawn uniformly from $\alpha t_{\rm cov}$02. The uncovered region is

$\alpha t_{\rm cov}$03

The geometric framework is built on positive reach and good partitions. For a closed set $\alpha t_{\rm cov}$04, one defines

$\alpha t_{\rm cov}$05

$\alpha t_{\rm cov}$06

A finite measurable partition

$\alpha t_{\rm cov}$07

is a good $\alpha t_{\rm cov}$08-partition if $\alpha t_{\rm cov}$09 and $\alpha t_{\rm cov}$10 for all $\alpha t_{\rm cov}$11.

If $\alpha t_{\rm cov}$12 has positive reach $\alpha t_{\rm cov}$13, then for every $\alpha t_{\rm cov}$14 there is a finite partition $\alpha t_{\rm cov}$15 of $\alpha t_{\rm cov}$16 with

$\alpha t_{\rm cov}$17

and therefore

$\alpha t_{\rm cov}$18

If $\alpha t_{\rm cov}$19 where $\alpha t_{\rm cov}$20 and $\alpha t_{\rm cov}$21, then one similarly gets

$\alpha t_{\rm cov}$22

and

$\alpha t_{\rm cov}$23

The proof proceeds by constructing a Whitney-type decomposition of the $\alpha t_{\rm cov}$24-interior

$\alpha t_{\rm cov}$25

using dyadic cubes of side $\alpha t_{\rm cov}$26. Cubes intersecting $\alpha t_{\rm cov}$27 lie entirely inside $\alpha t_{\rm cov}$28, have diameter at most $\alpha t_{\rm cov}$29, and volume $\alpha t_{\rm cov}$30. Fattening by $\alpha t_{\rm cov}$31 yields overlapping patches of diameter at most $\alpha t_{\rm cov}$32, 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 $\alpha t_{\rm cov}$33 does not have positive reach, the multiscale flat norm provides a surrogate. If $\alpha t_{\rm cov}$34 has finite perimeter, $\alpha t_{\rm cov}$35, and $\alpha t_{\rm cov}$36 satisfies

$\alpha t_{\rm cov}$37

where $\alpha t_{\rm cov}$38 is a minimizer of $\alpha t_{\rm cov}$39, then the denoised set $\alpha t_{\rm cov}$40 has reach $\alpha t_{\rm cov}$41 and admits a partition of cells of diameter $\alpha t_{\rm cov}$42 and measure $\alpha t_{\rm cov}$43. Intersecting those cells with $\alpha t_{\rm cov}$44 yields an $\alpha t_{\rm cov}$45-almost-partition with $\alpha t_{\rm cov}$46, and hence

$\alpha t_{\rm cov}$47

These results make explicit how coverage depends on $\alpha t_{\rm cov}$48, $\alpha t_{\rm cov}$49, and the geometry of $\alpha t_{\rm cov}$50, and they provide exponentially small tail bounds for failure of complete or almost-complete coverage (Alvarado et al., 2021).

5. Uncovered density in arithmetic covering systems

A covering system is a finite family of arithmetic progressions

$\alpha t_{\rm cov}$51

whose union is all of $\alpha t_{\rm cov}$52. The uncovered set is

$\alpha t_{\rm cov}$53

and its asymptotic density is

$\alpha t_{\rm cov}$54

The main theorem gives a sharp sufficient condition ensuring that $\alpha t_{\rm cov}$55 stays bounded away from $\alpha t_{\rm cov}$56. The paper introduces a mildly growing multiplicative weight $\alpha t_{\rm cov}$57 on the moduli and sets

$\alpha t_{\rm cov}$58

For any $\alpha t_{\rm cov}$59, there is $\alpha t_{\rm cov}$60 such that if the moduli are distinct, satisfy $\alpha t_{\rm cov}$61, and obey $\alpha t_{\rm cov}$62, then

$\alpha t_{\rm cov}$63

When the moduli lie in $\alpha t_{\rm cov}$64 with $\alpha t_{\rm cov}$65, this yields

$\alpha t_{\rm cov}$66

confirming the Erdős--Graham conjecture for moduli in $\alpha t_{\rm cov}$67.

The proof is a sieve in stages. Writing $\alpha t_{\rm cov}$68 and listing its prime divisors $\alpha t_{\rm cov}$69, one exposes the congruence classes prime by prime. If $\alpha t_{\rm cov}$70 is the set of integers not yet covered after stage $\alpha t_{\rm cov}$71, and $\alpha t_{\rm cov}$72 is the fraction of the $\alpha t_{\rm cov}$73-fiber at $\alpha t_{\rm cov}$74 that is removed at stage $\alpha t_{\rm cov}$75, then the first--second moment lemma bounds the removed mass by

$\alpha t_{\rm cov}$76

A change-of-measure argument then yields an explicit lower bound on $\alpha t_{\rm cov}$77, from which the estimate $\alpha t_{\rm cov}$78 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 $\alpha t_{\rm cov}$79 to

$\alpha t_{\rm cov}$80

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 $\alpha t_{\rm cov}$81 or $\alpha t_{\rm cov}$82 (Balister et al., 2018).

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 $\alpha t_{\rm cov}$83, 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 (Olesker-Taylor et al., 2019, Chen et al., 17 Nov 2025, Alvarado et al., 2021, Balister et al., 2018).

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 (Tan, 12 Nov 2025).