Epsilon Nets in Combinatorial Geometry

• Epsilon nets are fundamental structures in combinatorial geometry and learning theory that guarantee every sufficiently large range within a set is intersected by a small auxiliary set.
• They are widely applied in geometric discrepancy, range searching, and PAC learning, with size bounds often determined by the VC-dimension of the range space.
• Recent advances include efficient deterministic constructions and generalizations such as weak, weighted, and ε‑t‑nets which enhance both theory and algorithmic performance.

An ε-net is a fundamental structure in combinatorial geometry and learning theory, ensuring that all "large" subsets (ranges) of a given set are intersected ("stabbed") by a small auxiliary set. ε-nets play a pivotal role across geometric discrepancy theory, range searching, statistical learning (PAC theory), and extremal combinatorics, with key connections to VC-dimension, approximation algorithms, and the structure of geometric set systems. Their theory includes both classical "strong" ε-nets (hitting points within the original ground set) and "weak" ε-nets (allowing arbitrary stabbing points), as well as significant geometric, algorithmic, and extremal consequences.

1. Formal Definitions and Variants

Let $(X, \mathcal{R})$ be a finite range space, with $X$ a ground set, $\mathcal{R}\subseteq 2^X$ a family of ranges.

• Strong ε-net: For $0 < \varepsilon < 1$ and finite $X$ of size $n$, a subset $N\subseteq X$ is an ε-net if

$$\forall R\in\mathcal{R} : |R\cap X| \ge \varepsilon n \implies N\cap R\neq\emptyset.$$

• Weak ε-net: For geometric contexts (e.g., convex sets in $\mathbb{R}^d$), a set $N \subset \mathbb{R}^d$ (not necessarily contained in $X$) is a weak ε-net for $(X, \mathcal R)$ if every $R\in\mathcal{R}$ with $|R\cap X|\ge \varepsilon n$ intersects $N$.
• Weighted ε-net: For weighted points or when fractional approximations are required, a set $N$, possibly with multiplicities/weights, is considered, and the ε-net condition demands intersection in a weighted sense (Bertschinger et al., 2020).
• ε-$t$-net: Generalizes the classical case: instead of stabbing with single points, an ε-$t$-net is a family of size-$t$ subsets such that every large range contains at least one such $t$-tuple (Alon et al., 2020).

2. Fundamental Results: Bounds, Constructions, and Complexity

VC-Dimension and Size Bounds

The combinatorial richness of $\mathcal{R}$ is measured by the VC-dimension $d$. The foundational theorem (Haussler–Welzl):

$$\text{If VC-dim}(\mathcal{R}) = d, \text{ then every } (X, \mathcal{R}) \text{ admits an } \varepsilon\text{-net of size } O\left(\frac{d}{\varepsilon}\,\log\frac{1}{\varepsilon}\right).$$

Equality holds up to constants: in general, there exist range spaces of VC-dimension $d \ge 2$ where every $\varepsilon$-net has size at least $\Omega((d/\varepsilon)\log(1/\varepsilon))$ (Pach et al., 2010, Mustafa et al., 2017).

For geometric range spaces:

• Halfspaces in $\mathbb{R}^2, \mathbb{R}^3$, disks in $\mathbb{R}^2$: The $O(1/\varepsilon)$ bound is achievable, omitting the logarithmic factor (Har-Peled et al., 2014, Bus et al., 2015).
• Rectangles in the plane: The tight bound is $O(\frac{1}{\varepsilon} \log\log(1/\varepsilon))$, and this is sharp (Pach et al., 2010).
• General convex sets in $\mathbb{R}^d$: VC-dimension is unbounded; strong $\varepsilon$-nets can be linear in $n$, but weak $\varepsilon$-nets of subexponential size in $1/\varepsilon$ exist (Rubin, 2021).

Lower Bounds

• For bounded VC-dimension, the logarithmic overhead is necessary (Pach et al., 2010).
• For weak nets and convex sets in $\mathbb{R}^d$, the best known lower bound is $$\Omega(\varepsilon^{-1} \log^{d-1} (1/\varepsilon))$$ (0812.5039).

Small Strong ε-nets

• Existence and sharp values for net size versus $\varepsilon$ are established for boxes/rectangles, halfspaces, and disks in low dimensions, with precise staircase behavior for rectangles in the plane (Ashok et al., 2012).

Algorithmic Constructions

Deterministic constructions almost match random sampling. For disks in the plane, an algorithm yields nets of size at most $13.4/\varepsilon$ using Delaunay triangulation and recursive partitioning; practical performance is better (≈$9/\varepsilon$) (Bus et al., 2015).

3. Weak ε-Nets: Structure, Bounds, and Geometric Complexity

For convex ranges (unbounded VC-dimension):

• Weak $\varepsilon$-nets of subexponential size exist in fixed dimension: $O^*\left( \varepsilon^{-\alpha_d-\gamma} \right)$ with $\alpha_d < d$ for all $d\geq 3$; specifically, $\alpha_3 = 2.558$, $\alpha_4 = 3.48$, and $\alpha_d \sim d - 1/2$ for large $d$ (Rubin, 2021).
• In the plane, the best upper bound is $O(\varepsilon^{-3/2-\gamma})$, improving the classical $O(\varepsilon^{-2})$ (Rubin, 2018).
• Lower bounds via the "stretched grid" construction indicate a superlinear dependency on $\varepsilon^{-1}$: $$\Omega(\varepsilon^{-1} \log^{d-1} (1/\varepsilon))$$ (0812.5039).

Hardness:

• Verifying a weak ε-net for convex sets in $\mathbb{R}^3$ is co-NP-hard (Knauer et al., 2011).

Positive-fraction intersection theorems enable $O(\varepsilon^{-2})$-size weak ε-nets for pairwise-induced families such as diametral balls and axis-aligned boxes, independent of $d$ (Magazinov et al., 2015).

4. Extensions: Weighted, t-Set, and Other Generalizations

Weighted ε-nets

Weighted ε-nets interpolate between ε-nets and ε-approximations. For size-2 nets for convex sets (with thresholds $\alpha_1, \alpha_2$), sharp trade-offs are obtained—e.g., in $\mathbb{R}^2$, $\alpha_1 \geq 4/7$ is tight (Bertschinger et al., 2020). Similar explicit results exist for axis-parallel boxes.

ε-$t$-nets and Extremal Applications

Generalized ε-nets with $t$-tuples (ε-$t$-nets) mirror classical Turán-type extremal problems, including Zarankiewicz's problem on $K_{t,t}$-free graphs. For hypergraphs of bounded VC-dimension, an ε-$t$-net of size $$O((1+\log t)d/\varepsilon \cdot \log(1/\varepsilon))$$ always exists (Alon et al., 2020, Keller et al., 2023). These structures lead to new proofs and sharp bounds for geometric incidence graphs.

5. Quantum, Metric, and Algorithmic Applications

• Quantum computing: ε-nets for unitary groups PU$(d)$ relate to approximate $t$-designs. New heat kernel–based results show that a $\delta$-approximate $t$-design is an ε-net for $\delta \gtrsim (\varepsilon/\sqrt{d})^{d^2}$, allowing more efficient quantum protocol design (Słowik et al., 11 Mar 2025, Oszmaniec et al., 2020).
• Metric embeddings, distance oracles: ε-nets for shortest-path set-systems (VC-dimension 2) yield small hitting sets and nearly optimal oracle/data-structure space for large-distance queries (Razenshteyn, 2012).
• Transversal/Helly-type: Tverberg-type theorems reinterpreted as weak ε-net statements facilitate new partition theorems for large convex intersections, with minimal dependence on ambient dimension (Soberón, 2017).
• Algorithmic geometric optimization: The size of ε-nets governs approximation ratios for geometric hitting set and set cover; improvements in ε-net bounds directly translate into improved algorithmic guarantees (Bus et al., 2015).

6. Geometry-Dependent Improvements and Parameter Hierarchies

• Under refined measures (shallow-cell complexity, Alexander's capacity, doubling constants), ε-net sizes can be much smaller than given by VC-dimension alone; $O(\log(1/\varepsilon))$ or even $O(1)$-size nets are possible for families with low complexity (Kupavskii et al., 2017).
• For disks, halfspaces (in $\mathbb{R}^2$ or $\mathbb{R}^3$), and pseudo-disks, optimal or near-optimal constant-factor nets exist (Har-Peled et al., 2014, Bus et al., 2015).
• For axis-parallel rectangles, the optimal bound is $O((1/\varepsilon)\log\log(1/\varepsilon))$ (Pach et al., 2010, Kupavskii et al., 2017).

7. Open Problems and Research Directions

• Closing the gap for weak ε-nets for convex sets: Is $O(1/\varepsilon)$ achievable in fixed dimension?
• Determining the correct exponent for the plane ($d=2$): current best $O(\varepsilon^{-3/2-\gamma})$ versus lower bounds.
• Extending hardness results for weak nets to other geometric range families and higher dimension (Knauer et al., 2011).
• Designing faster, practical algorithms for constructing weak ε-nets with nearly optimal size.
• Developing ε-$t$-net theory for more complex settings and additional extremal graph applications.
• Understanding the complexity of weighted ε-nets and their role in robust geometric approximation.

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References:

• "Tighter Estimates for epsilon-nets for Disks" (Bus et al., 2015)
• "Stronger Bounds for Weak Epsilon-Nets in Higher Dimensions" (Rubin, 2021)
• "An Improved Bound for Weak Epsilon-Nets in the Plane" (Rubin, 2018)
• "Lower bounds for weak epsilon-nets and stair-convexity" (0812.5039)
• "Tight lower bounds for the size of epsilon-nets" (Pach et al., 2010)
• "When are epsilon-nets small?" (Kupavskii et al., 2017)
• "Small Strong Epsilon Nets" (Ashok et al., 2012)
• "Weighted Epsilon-Nets" (Bertschinger et al., 2020)
• "The $ε$-$t$-Net Problem" (Alon et al., 2020)
• "Zarankiewicz's problem via $ε$-t-nets" (Keller et al., 2023)
• "Epsilon-Nets for Halfspaces Revisited" (Har-Peled et al., 2014)
• "On Epsilon-Nets, Distance Oracles, and Metric Embeddings" (Razenshteyn, 2012)
• "Tverberg partitions as weak epsilon-nets" (Soberón, 2017)
• "Positive-fraction intersection results and variations of weak epsilon-nets" (Magazinov et al., 2015)
• "Epsilon-nets, unitary designs and random quantum circuits" (Oszmaniec et al., 2020)
• "Fundamental solutions of heat equation on unitary groups establish an improved relation between $ε$-nets and approximate unitary $t$-designs" (Słowik et al., 11 Mar 2025)