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world:boule [2025/10/03 01:56] rdouc created |
world:boule [2025/10/03 09:06] (current) rdouc [Proof] |
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| - | **Question** | ||
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| + | ====== Question ====== | ||
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| + | <WRAP center round tip 80%> | ||
| Let $D$ be an infinite set without accumulation points in $\mathbb{R}^d$. | Let $D$ be an infinite set without accumulation points in $\mathbb{R}^d$. | ||
| Show that, for every $n \in \mathbb{N}^*$, there exists a ball containing exactly $n$ points of $D$. | Show that, for every $n \in \mathbb{N}^*$, there exists a ball containing exactly $n$ points of $D$. | ||
| - | **Proof** | + | </WRAP> |
| - | First, note that $D$ must be countable. Indeed, for each point $x \in D$, one can associate a *rational ball* (that is, a ball with rational center coordinates and rational radius) containing $x$ as the only element of $D$. This defines an injection from $D$ into the countable set of rational balls, which proves that $D$ is countable. | + | ===== Proof ===== |
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| + | First, note that $D$ must be countable. Indeed, for each point $x \in D$, one can associate a **rational ball** (that is, a ball with rational center coordinates and rational radius) containing $x$ as the only element of $D$. This defines an injection from $D$ into the countable set of rational balls, which proves that $D$ is countable. | ||
| Now consider, for $(a,b) \in D \times D$, the affine hyperplane | Now consider, for $(a,b) \in D \times D$, the affine hyperplane | ||
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| $$ | $$ | ||
| - | To order these distances strictly, it is enough to check that, for every $r>0$, the set $A \cap [0,r]$ is finite. But $A \cap [0,r]$ corresponds to the number of points in $D \cap B(c,r)$, which is a compact set containing no accumulation points. Hence it is finite. | + | To order these distances strictly, it is now enough to check that, for every $r>0$, the set $A \cap [0,r]$ is finite. But the cardinal of $A \cap [0,r]$ corresponds to the number of points in $D \cap \overline{B(c,r)}$, which is a compact set containing no accumulation points. Hence it is finite. |
| - | We can thus write $A = \{\rho_n \; ; \; n \geq 1\}$ with $\rho_n$ strictly increasing. | + | We can thus write $A = \{\rho_n \; ; \; n \geq 1\}$ with strictly increasing $(\rho_n)$. |
| For every $n \geq 1$, by choosing a radius $\rho$ such that $\rho_n < \rho < \rho_{n+1}$, the ball $B(c,\rho)$ contains exactly $n$ points of $D$. | For every $n \geq 1$, by choosing a radius $\rho$ such that $\rho_n < \rho < \rho_{n+1}$, the ball $B(c,\rho)$ contains exactly $n$ points of $D$. | ||
| ---- | ---- | ||
| - | **Remarks:** | ||
| - | * One can avoid Baire’s theorem by using a measure-theoretic argument. Let $\lambda_d$ be the Lebesgue measure on $\mathbb{R}^d$. Since $\lambda_d(H_{a,b})=0$, we get $\lambda_d(\bigcup_{a,b \in D} H_{a,b})=0$. Hence $\bigcup_{a,b \in D} H_{a,b} \neq \mathbb{R}^d$. | + | <WRAP center round important 80%> |
| + | ===== Remarks: ===== | ||
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| + | * One can avoid Baire’s theorem by using a measure-theoretic argument. Let $\lambda_d$ be the Lebesgue measure on $\mathbb{R}^d$. Since $\lambda_d(H_{a,b})=0$ and $D$ is countable, we get $\lambda_d(\bigcup_{a,b \in D} H_{a,b})=0$. Hence $\bigcup_{a,b \in D} H_{a,b} \neq \mathbb{R}^d$. | ||
| * For $D = \mathbb{Z}^d$, one can construct $c$ explicitly, for example $c=(\pi,\ldots,\pi^d)$. Then, for $a,b \in \mathbb{Z}^d$, the equality $\|c-a\|^2=\|c-b\|^2$ becomes a polynomial equation with integer coefficients having $\pi$ as a root. This is impossible unless $a=b$, since $\pi$ is not algebraic. However, this explicit construction of $c$ does not generalize (or at least not simply) when $D$ is not $\mathbb{Z}^d$. | * For $D = \mathbb{Z}^d$, one can construct $c$ explicitly, for example $c=(\pi,\ldots,\pi^d)$. Then, for $a,b \in \mathbb{Z}^d$, the equality $\|c-a\|^2=\|c-b\|^2$ becomes a polynomial equation with integer coefficients having $\pi$ as a root. This is impossible unless $a=b$, since $\pi$ is not algebraic. However, this explicit construction of $c$ does not generalize (or at least not simply) when $D$ is not $\mathbb{Z}^d$. | ||
| * Obviously, when $D = \mathbb{Z}^d$, one does not need to prove that $D$ is countable, nor that $A \cap [0,r]$ is finite, which significantly shortens the proof. | * Obviously, when $D = \mathbb{Z}^d$, one does not need to prove that $D$ is countable, nor that $A \cap [0,r]$ is finite, which significantly shortens the proof. | ||
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| + | </WRAP> | ||
world/boule.1759449407.txt.gz · Last modified: 2025/10/03 01:56 by rdouc
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