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$\newcommand\C{\mathscr C}\newcommand\ep{\varepsilon}$Let $\C$ denote the set of all disks of a radius $r\in(0,\infty)$ contained in the unit square. Using a rectangular grid of centers of disks in $\C$, we can cover $\C$ by $N=O(r^2/\ep^2)$ balls of radius $\ep\in(0,\infty)$, where the balls are considered with respect to the distance between disks equal the Lebesgue measure of the symmetric difference between the disks and the constant in $O(\cdot)$ is universal.

Hence, Talagrand's Theorem 1.1 for empirical measures (used here with $v=2$), implies the following: $$P\Big(\sup_{C\in\C}\Big|\sum_1^n 1(p_j\in C)-\pi r^2 n\Big|\ge z\sqrt n\Big) \le K(r)z^3e^{-2z^2} \tag{1}$$ for all real $z>0$, where $K(r)\in(0,\infty)$ depends only on $r$ (in a way that can be found from the proof).

Obviously, a condition of the form $z\ge c$ with a universal constant $c\in(0,\infty)$ is missing here; cf. e.g. Azencott and Vayatis, p. 564.

$\newcommand\C{\mathscr C}\newcommand\ep{\varepsilon}$Let $\C$ denote the set of all disks of a radius $r\in(0,\infty)$ contained in the unit square. Using a rectangular grid of centers of disks in $\C$, we can cover $\C$ by $N=O(r^2/\ep^2)$ balls of radius $\ep\in(0,\infty)$, where the balls are considered with respect to the distance between disks equal the Lebesgue measure of the symmetric difference between the disks and the constant in $O(\cdot)$ is universal.

Hence, Talagrand's Theorem 1.1 for empirical measures (used here with $v=2$) implies the following: $$P\Big(\sup_{C\in\C}\Big|\sum_1^n 1(p_j\in C)-\pi r^2 n\Big|\ge z\sqrt n\Big) \le K(r)z^3e^{-2z^2} \tag{1}$$ for all real $z>0$, where $K(r)\in(0,\infty)$ depends only on $r$ (in a way that can be found from the proof).

Obviously, a condition of the form $z\ge c$ with a universal constant $c\in(0,\infty)$ is missing here; cf. e.g. Azencott and Vayatis, p. 564.

$\newcommand\C{\mathscr C}\newcommand\ep{\varepsilon}$Let $\C$ denote the set of all disks of a radius $r\in(0,\infty)$ contained in the unit square. Using a rectangular grid of centers of disks in $\C$, we can cover $\C$ by $N=O(r^2/\ep^2)$ balls of radius $\ep\in(0,\infty)$, where the balls are considered with respect to the distance between disks equal the Lebesgue measure of the symmetric difference between the disks and the constant in $O(\cdot)$ is universal.

Hence, Talagrand's Theorem 1.1 for empirical measures (use here with $v=2$, implies the following: $$P\Big(\sup_{C\in\C}\Big|\sum_1^n 1(p_j\in C)-\pi r^2 n\Big|\ge z\sqrt n\Big) \le K(r)z^3e^{-2z^2} \tag{1}$$ for all real $z>0$, where $K(r)\in(0,\infty)$ depends only on $r$ (in a way that can be found from the proof).

Obviously, a condition of the form $z\ge c$ with a universal constant $c\in(0,\infty)$ is missing here; cf. e.g. Azencott and Vayatis, p. 564.

$\newcommand\C{\mathscr C}\newcommand\ep{\varepsilon}$Let $\C$ denote the set of all disks of a radius $r\in(0,\infty)$ contained in the unit square. Using a rectangular grid of centers of disks in $\C$, we can cover $\C$ by $N=O(r^2/\ep^2)$ balls of radius $\ep\in(0,\infty)$, where the balls are considered with respect to the distance between disks equal the Lebesgue measure of the symmetric difference between the disks and the constant in $O(\cdot)$ is universal.

Hence, Talagrand's Theorem 1.1 for empirical measures (used here with $v=2$), implies the following: $$P\Big(\sup_{C\in\C}\Big|\sum_1^n 1(p_j\in C)-\pi r^2 n\Big|\ge z\sqrt n\Big) \le K(r)z^3e^{-2z^2} \tag{1}$$ for all real $z>0$, where $K(r)\in(0,\infty)$ depends only on $r$ (in a way that can be found from the proof).

Obviously, a condition of the form $z\ge c$ with a universal constant $c\in(0,\infty)$ is missing here; cf. e.g. Azencott and Vayatis, p. 564.

$\newcommand\C{\mathscr C}\newcommand\ep{\varepsilon}$Let $\C$ denote the set of all disks of a radius $r\in(0,\infty)$ contained in the unit square. Using a rectangular grid of centers of disks in $\C$, we can cover $\C$ by $N=O(r^2/\ep^2)$ balls of radius $\ep\in(0,\infty)$, where the balls are considered with respect to the distance between disks equal the Lebesgue measure of the symmetric difference between the disks and the constant in $O(\cdot)$ is universal.

Hence, Talagrand's Theorem 1.1 for empirical measures (use here with $v=2$, implies the following: $$P\Big(\sup_{C\in\C}\Big|\sum_1^n 1(p_j\in C)-\pi r^2 n\Big|\ge z\sqrt n\Big) \le K(r)z^3e^{-2z^2} \tag{1}$$ for all real $z>0$, where $K(r)\in(0,\infty)$ depends only on $r$ (in a way that can be found from the proof).

Obviously, a condition of form $z\ge c$ with a universal constant $c\in(0,\infty)$ is missing here; cf. e.g. Azencott and Vayatis, p. 564.

$\newcommand\C{\mathscr C}\newcommand\ep{\varepsilon}$Let $\C$ denote the set of all disks of a radius $r\in(0,\infty)$ contained in the unit square. Using a rectangular grid of centers of disks in $\C$, we can cover $\C$ by $N=O(r^2/\ep^2)$ balls of radius $\ep\in(0,\infty)$, where the balls are considered with respect to the distance between disks equal the Lebesgue measure of the symmetric difference between the disks and the constant in $O(\cdot)$ is universal.

Hence, Talagrand's Theorem 1.1 for empirical measures (use here with $v=2$, implies the following: $$P\Big(\sup_{C\in\C}\Big|\sum_1^n 1(p_j\in C)-\pi r^2 n\Big|\ge z\sqrt n\Big) \le K(r)z^3e^{-2z^2} \tag{1}$$ for all real $z>0$, where $K(r)\in(0,\infty)$ depends only on $r$ (in a way that can be found from the proof).

Obviously, a condition of the form $z\ge c$ with a universal constant $c\in(0,\infty)$ is missing here; cf. e.g. Azencott and Vayatis, p. 564.