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Let $C$ be the square $[-1,1]^2$. Let $a_1,\dots,a_m$ be points chosen independently and uniformly at random from $C$. Let $d_m$ (dispersion) be the random variable $\max_{x \in C}{\min_{j \in [m]}{\|x-a_j\|_2}}$. I would like to find a function $\epsilon(m)$ such that $\Pr[d_m \leq \epsilon(m)] \geq $ constant (no dependence on m).

In computational geometry terminology, this means, find an $\epsilon(m)$ such that a "random" $m$-subset of $C$ is an $\epsilon(m)$-net for $(C,\mathcal{B})$ (where $\mathcal{B}$ is the class of all $L_2$-balls in $\mathbb{R}^2$) with constant probability.

In the one-dimensional case (here $C = [-1,1]$), we can take $\epsilon(m)$ to be $K\frac{\log m}{m}$ for a large enough constant $K$. This follows from Theorem 3.1 of "On the lengths of the pieces of a stick broken at random (Holst, 1980)".

I found this similar question on mathoverflow, about the expected value of maximum distance between any two points in $\{a_1,\dots,a_m\}$. I don't see how the bound there can be used to get an $\epsilon(m)$ though.

On page 2 of the paper "Quasi-Monte-Carlo methods and the dispersion of point sequences (Rote-Tichy, 1996)" it says that "If the range space $R$ has finite Vapnik-Chervonenkis dimension $d$ then a random subset of $X$ [to be thought of as $C$ in our question] of size $(d/\epsilon)\log (1/\epsilon)$ is an $\epsilon$-net with high probability." This would be great for us as $\mathcal{B}$ (the class of all $L_2$-balls in $\mathbb{R}^2$) has finite VC dimension ($= d+1$ for the $d$-dimensional case). Rote-Tichy says that this is proved in the paper "Epsilon-nets and simplex range queries (Haussler-Welzl, 1987)". But on this paper, I could only find such theorems for random subsets of a finite $A \subset X$ (Theorem 3.3, Corollary 3.7, etc.).

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  • $\begingroup$ One should be careful: The $\epsilon$ in the Rote-Tichy paper is the measure, i.e. the area of the circle (or volume of the ball in $d$ dimensions), whereas the $\epsilon$ in the Question is the distance, i.e. the radius of the circle. Thus, the claim from the Rote-Tichy paper (and from $\epsilon$-net theory) coincides with that from the Answer. $\endgroup$ Commented May 3, 2023 at 23:41
  • $\begingroup$ The original Haussler-Welzl paper from 1987 may not contain the general statement, but by now this is in monographs and textbooks. See for example Nabil H. Mustafa, Sampling in Combinatorial and Geometric Set Systems, vol. 265 of Mathematical Surveys and Monographs. American Mathematical Society,2022.doi:10.1090/surv/265. $\endgroup$ Commented May 3, 2023 at 23:44

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For every constant dimension $d$, (I.e. when $C = [0,1]^d$), the answer is asymptotically $\varepsilon_d(m) = \Theta\left(\frac{\log m}{m}\right)^{1/d}$. In the $\Theta$ notation I’m hiding constant factors that depend on $d$. This follows from the coupon collectors problem: let us partition a $[0,1]^d$ cube into $\varepsilon^{-d}$ sub-cubes of side-length $\varepsilon$, I.e. $[0,1]^d = \left( \bigcup_i [i \varepsilon, (i + 1) \varepsilon]\right)^d$.

Now, let us draw $m$ points uniformly at random from $[0, 1]^d$, and let’s keep track of which sub-cubes are being hit.

By coupon collectors, as soon as $m \gg \varepsilon^{-d} \log(\varepsilon^{-d})$, with high probability we will hit each sub-cube, and when $m \ll \varepsilon^{-d} \log(\varepsilon^{-d})$ with high probability we will miss at least one sub-cube.

Now it is enough to show the following two elementary geometric fact:

  1. Any subset $S\subset [0, 1]^d$ which misses at least one sub-cube, has dispersion at least $\varepsilon/2$ (take the center of a missed cube as a witness $x$).

  2. Any subset $S \subset [0,1]^d$ which hits every sub-cube has dispersion at most $\sqrt{d}\varepsilon$ (the diameter of a cube of side length $\varepsilon$).

Related: I recommend the great answer by Iosif Pinelis with much more precise calculation of the exact constant for the case where $d=1$.

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