Dvoretzky Random Covering

• Dvoretzky Random Covering is a probabilistic covering problem where randomly placed sets, like arcs or balls, are used to study full coverage conditions and fractal exceptional sets.
• The methodology leverages harmonic analysis and Fourier techniques to derive necessary and sufficient criteria for almost sure complete coverage of geometric spaces.
• Extensions include non-uniform distributions, higher-dimensional analogues, and dynamical variants, revealing critical phase transitions and connections to metric geometry and topology.

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 $c < 1$ (Hirayama et al., 2021).

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 (Hirayama et al., 2021, Fan et al., 2019) 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 $f$ (Fan et al., 2019).

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 (Hirayama et al., 2021).

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 (Chen et al., 17 Nov 2025, Hirayama et al., 2021). 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,

$$\dim_{\mathcal{F}} K_\ell = \dim_{\mathcal{H}} K_\ell = 1 - D_\ell$$

(Chen et al., 17 Nov 2025). 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 (Tan, 12 Nov 2025).

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 (Järvenpää et al., 2015, Li et al., 2013). 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

$$\dim_H E = a, \quad a = \limsup_{n\to\infty} \frac{\log n}{-\log r_n}$$

(Li et al., 2013). 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 (Järvenpää et al., 2015, Li et al., 2013).

A table summarizing some key thresholds:

Context	Coverage Threshold	Reference
Circle, uniform, $\ell_n=c/n$	$c > 1$ full cover; $c<1$ fails	(Hirayama et al., 2021)
Circle, density $f$, $\ell_n=c/n$	$c m_f \geq 1$ full cover	(Fan et al., 2019)
Sphere, dimension $d\to\infty$	Max coverage $1-e^{-1}$ near-deterministic	(Hoehner et al., 17 Jan 2025)

In higher dimensions, randomness leads to asymptotically optimal sphere coverings (coverage ratio $1-e^{-\rho}$ at total density $\rho=1$) (Hoehner et al., 17 Jan 2025).

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 (Koivusalo et al., 2021). 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 (Barany et al., 23 Jun 2025). 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 (Tan, 12 Nov 2025).

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 (Chen et al., 17 Nov 2025). 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 (Komendarczyk et al., 2012). 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.

• (Hirayama et al., 2021) On genericity of non-uniform Dvoretzky coverings of the circle.
• (Tan, 12 Nov 2025) The non-covered set in Dvoretzky covering is a set of multiplicity.
• (Chen et al., 17 Nov 2025) Salem properties of Dvoretzky random coverings.
• (Fan et al., 2019) On $\mu$-Dvoretzky random covering of the circle.
• (Hoehner et al., 17 Jan 2025) On the Optimality of Random Partial Sphere Coverings in High Dimensions.
• (Koivusalo et al., 2021) Uniform random covering problems.
• (Li et al., 2013) A note on the hitting probabilities of random covering sets.
• (Järvenpää et al., 2015) Hitting probabilities of random covering sets in tori and metric spaces.
• (Komendarczyk et al., 2012) Finite random coverings of one-complexes and the Euler characteristic.
• (Barany et al., 23 Jun 2025) Dynamical covering sets in self-similar sets.