---
title: Dvoretzky Random Covering
url: https://www.emergentmind.com/topics/dvortezky-random-covering
type: topic
---

# Dvoretzky Random Covering

Dvoretzky random covering describes a class of probabilistic covering problems, with origins in geometric probability and harmonic analysis, in which randomly placed sets (typically arcs or balls) are used to cover a geometric space such as the circle or the sphere. The fundamental question is to characterize, in terms of the underlying geometry and stochastic process, the conditions under which these randomly placed sets cover every point (or “almost every” point) of the space with probability one, as well as the properties of the (potentially fractal) exceptional sets that remain uncovered. The theory connects limit theorems, fractal geometry, Fourier analysis, and aspects of random processes. The prototypical problem—Dvoretzky’s random covering of the circle—has produced influential criteria, phase transitions, and deep links with uniqueness problems for trigonometric series and the metric geometry of random sets.

## 1. Formulation and Classical Results

Consider the unit circle $\mathbb{T} = \mathbb{R}/\mathbb{Z}$ and a deterministic sequence of lengths $\{\ell_n\}_{n\ge 1}$ with $0 < \ell_n < 1$. Let $\omega_n$ be i.i.d. random variables, each uniformly distributed on $\mathbb{T}$. Form the open arcs $I_n(\omega) = (\omega_n-\ell_n/2,\,\omega_n+\ell_n/2)\;\mathrm{mod}\;1$. The random limsup set is 
$$
C(\omega) = \limsup_{n\to\infty} I_n(\omega) = \{ x\in \mathbb{T}: x\in I_n(\omega) \text{ for infinitely many } n \}.
$$
Dvoretzky’s problem asks for a necessary and sufficient criterion on $\{\ell_n\}$ for $C(\omega) = \mathbb{T}$ almost surely, that is, for the random arcs to cover every point of the circle infinitely often with probability one.

By the second Borel–Cantelli lemma, $\lambda$-almost every point is covered infinitely often if and only if $\sum_n \ell_n = \infty$. However, full coverage (in the sense that every point is covered infinitely often, almost surely) requires a subtler condition. The solution, due to Shepp (1972), is:

$$
C(\omega)=\mathbb{T} \text{ a.s.} \Longleftrightarrow \sum_{n=1}^\infty \frac{1}{n^2\, e^{S_n}} = \infty,\quad S_n = \ell_1+\cdots+\ell_n.
$$

For the special case $\ell_n = c/n$, the threshold is at $c=1$: full coverage occurs for $c > 1$, fails for $c < 1$ [2110.07350].

## 2. Extensions: Non-Uniform Dvoretzky Coverings

A major extension replaces the uniform distribution of the center points $\omega_n$ with an absolutely continuous probability measure $\mu_f$ having density $f \in L^1(\mathbb{T})$. Define the essential infimum $m_f = \mathrm{ess\,inf}_{\mathbb{T}} f(x)$ and the set $K_f$ of points where $f$ attains $m_f$. The size of $K_f$ is quantified by its Hausdorff dimension and upper box dimension.

The principal sharp condition [2110.07350, 1910.08056] is as follows: suppose $m_f > 0$ and the upper box dimension of $K_f$ is $<1$. Then, defining $D = \limsup_{n\to\infty} (\ell_1+\cdots+\ell_n) / \log n$, 
$$
D \geq 1/m_f \quad \Longleftrightarrow \quad \mathbb{T} \text{ is a.s. fully covered}.
$$
Sufficiency can be extended if $K_f$ has Hausdorff dimension $<1$ and additional uniformity conditions on $\ell_n$ are imposed. In the special case $\ell_n = c/n$, the threshold is $cm_f \geq 1$—extending the classical case without any smoothness assumptions on the density $f$ [1910.08056].

Menshov-type genericity holds: if $|K_f|=0$ and the threshold is met, one can construct arbitrarily small perturbations of $f$ (differing on a set of Lebesgue measure $<\varepsilon$) for which coverage holds. This does not apply for the uniform density [2110.07350].

## 3. Fractal Structure and Multiplicative Chaos of the Uncovered Set

When the coverage threshold is not met, the complementary limsup set $E = \mathbb{T} \setminus C(\omega)$ forms a random closed set with fractal structure. Motivated by work in harmonic analysis, Dvoretzky-type random limsup sets have been analyzed in terms of their Hausdorff and Fourier (Salem) dimensions [2511.13068, 2110.07350]. Kahane’s Theorem gives:

$$
\dim_{\mathcal{H}} K_\ell = 1 - D_\ell,\;\; D_\ell = \limsup_{k\to\infty}\frac{\ell_1+\cdots+\ell_k}{\log k}
$$
almost surely when $K_\ell$ is nonempty ($C(\omega)\neq\mathbb{T}$).

Recent work establishes that $K_{\ell}$ is almost surely a Salem set in the subcritical regime: its Fourier dimension matches its Hausdorff dimension, that is,
$$
\dim_{\mathcal{F}} K_\ell = \dim_{\mathcal{H}} K_\ell = 1 - D_\ell
$$
[2511.13068]. The construction relies on a multiplicative chaos measure $\mu_{RC}$ supported on $K_\ell$, whose Fourier coefficients decay at the optimal rate.

Concurrently, the multiplicative chaos measure $\mu_D$ (the Dvoretzky measure) built from the survival martingales $M_n(t)$
$$
M_n(t) = \prod_{k=1}^n \frac{1 - 1_{I_k}(t)}{1-\ell_k}
$$
has the Rajchman property (Fourier coefficients tend to zero), and $K_\ell$ is a set of multiplicity in the sense of trigonometric series uniqueness theory [2511.09726].

## 4. Generalizations: Higher Dimensions and Metric Spaces

Analogous Dvoretzky-type random covering problems have been formulated on tori of higher dimension and on compact metric spaces with Ahlfors regular measure [1510.06630, 1307.2819]. For balls of radii $r_n \downarrow 0$ placed at i.i.d. random centers, the random limsup set $E = \limsup_{n} B(x_n, r_n)$ almost surely satisfies
$$
\dim_H E = a, \quad a = \limsup_{n\to\infty} \frac{\log n}{-\log r_n}
$$
[1307.2819]. For intersections with a fixed analytic set $F$, hitting-probability and intersection-dimension dichotomies are established: if $\dim_H F > d-a$, then $E \cap F \neq \emptyset$ almost surely; otherwise, intersection fails almost surely [1510.06630, 1307.2819].

A table summarizing some key thresholds:

| Context                      | Coverage Threshold                         | Reference          |
|------------------------------|--------------------------------------------|--------------------|
| Circle, uniform, $\ell_n=c/n$| $c > 1$ full cover; $c<1$ fails          | [2110.07350]       |
| Circle, density $f$, $\ell_n=c/n$ | $c m_f \geq 1$ full cover                 | [1910.08056]       |
| Sphere, dimension $d\to\infty$ | Max coverage $1-e^{-1}$ near-deterministic | [2501.10607]       |

In higher dimensions, randomness leads to asymptotically optimal sphere coverings (coverage ratio $1-e^{-\rho}$ at total density $\rho=1$) [2501.10607].

## 5. Uniform and Dynamical Covering Variants

“Uniform random covering” investigates liminf-type sets, i.e., the set of points eventually always covered by some random ball at each layer. It reveals a richer phase structure: full covering, full-measure covering, and countable exceptional sets, each governed by separate thresholds on the decay rates of the radii [2103.01595]. Unlike the classical limsup Dvoretzky problem, uniform covering thresholds involve probabilities of not being covered within a finite number of layers.

In fractal and dynamical contexts, “dynamical Dvoretzky covering” considers shrinking targets along orbits in self-similar sets with Bernoulli measures [2506.18447]. The critical covering behavior is characterized via a thermodynamic pressure function, with explicit phase transitions for full set covering, full measure, or positive-codimension sets, unifying dynamical covering and classical Borel–Cantelli arguments.

## 6. Connections to Harmonic Analysis: Uniqueness and Multiplicity

Dvoretzky random covering is closely linked to harmonic analysis, specifically the structure of uniqueness (U-sets) and multiplicity (M-sets) for trigonometric series. In the non-covering regime, the exceptional set is almost surely a set of multiplicity, as witnessed by the existence of measures with vanishing Fourier coefficients supported on the set [2511.09726].

For subcritical coverings, the corresponding uncovered sets are Salem sets: they achieve the maximal possible Fourier dimension (equal to Hausdorff dimension) for random fractals of their kind [2511.13068]. The Rajchman and Salem properties are established by controlling the Fourier decay of the associated multiplicative chaos measures.

## 7. Topological and Algebraic-Topological Perspective

The finite version of Dvoretzky random covering, especially on one-dimensional complexes, can be exactly analyzed using algebraic-topological invariants such as the nerve complex and Euler characteristic [1207.1133]. For a “good” random covering, full coverage of the space corresponds to the minimal value of the relative Euler characteristic of the nerve complex. In the circle case, the classical inclusion–exclusion formula for full coverage probability is recovered from this perspective, yielding a full combinatorial and probabilistic account of the model.

## References

- [2110.07350] On genericity of non-uniform Dvoretzky coverings of the circle.
- [2511.09726] The non-covered set in Dvoretzky covering is a set of multiplicity.
- [2511.13068] Salem properties of Dvoretzky random coverings.
- [1910.08056] On $\mu$-Dvoretzky random covering of the circle.
- [2501.10607] On the Optimality of Random Partial Sphere Coverings in High Dimensions.
- [2103.01595] Uniform random covering problems.
- [1307.2819] A note on the hitting probabilities of random covering sets.
- [1510.06630] Hitting probabilities of random covering sets in tori and metric spaces.
- [1207.1133] Finite random coverings of one-complexes and the Euler characteristic.
- [2506.18447] Dynamical covering sets in self-similar sets.

Source: https://www.emergentmind.com/topics/dvortezky-random-covering