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Please help me to construct an example of an injective map $F: \{0,1\}^{n} \to \mathbb{N}$ such that $$ |F(x_1,\ldots,x_n)-F(y_1,\ldots,y_n)| = \sum_{k=1}^{n} |x_k-y_k| $$ for all $x=(x_1,\ldots,x_n), \quad y=(y_1,\ldots,y_n) \quad (x_i, y_i =0 \,\,\text{or}\,\, 1)$.

Of course, there are $2^n$ such tuples and every such an $n$-tuple corresponds to a natural number.

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  • $\begingroup$ $\mathbb{N}_0$ or $\mathbb{N}_+$? $\endgroup$ – orlp Jun 12 '18 at 8:33
  • $\begingroup$ $\mathbb{N}=\{0,1,2,\ldots\}$ or $\mathbb{N}=\{1,2,\ldots\}$ - it does not matter $\endgroup$ – monica_k Jun 12 '18 at 8:35
  • $\begingroup$ I think maybe Gray codes can help here. $\endgroup$ – monica_k Jun 12 '18 at 8:38
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This is impossible. Consider $x$ to be a $n$-tuple with $n-1$ zeroes and a single $1$ in the $k$th position and $y = (0, \dots, 0)$.

Then:

$$|F(x_1, \dots, x_n) - F(0, \dots, 0)| = 1$$

This means that there are only two possible values for $F(x_1, \dots, x_n)$, namely $F(0, \dots, 0) \pm 1$.

But there are $n$ possible $x$ with the conditions above, meaning for $n > 2$ the mapping can't be injective.

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  • $\begingroup$ Thank you very much! And what will be if one replace the above identity in the problem with the inequality $$ |F(x_1,\ldots,x_n)-F(y_1,\ldots,y_n)| \leq C \sum_{k=1}^{n} |x_k-y_k| $$ for some $C>0$ ? $\endgroup$ – monica_k Jun 12 '18 at 8:42
  • $\begingroup$ @monica_k Well by the same argument as above, $C$ can not be a constant irrespective of $n$, as it needs to be at least $n/2$. I don't know if that would be sufficient, but it would at least be necessary. $\endgroup$ – orlp Jun 12 '18 at 8:43
  • $\begingroup$ @monica_k As there are $n$ choices for $\mathbf x$ (for given $\mathbf y$) that leadt to $\sum=1$, you'll need $C\ge n/2$. However, there are $n\choose 2$ possibilities with $\sum=2$, so you need to pack ${n\choose 2}+n+1$ values into $2C+1$ allowed values, i.e., $C$ must be ate least $\sim n^2/4$. We could continue, or note directly that to cover all vectros, we need $2^n\le 2nC+1$ $\endgroup$ – Hagen von Eitzen Jun 12 '18 at 8:47
  • $\begingroup$ I'm sorry. Aren't there any positive examples for similar problems? I try the last attempt: What about an example of an injective map $F: B_{n,m} \to \mathbb{N}$ such that $$ |F(x_1,\ldots,x_n)-F(y_1,\ldots,y_n)| \leq C \sum_{k=1}^{n} |x_k-y_k| $$ for all $x,y\in B_{n,m}$ (for some $C>0$) ? Here $$ B_{n, m}=\{x=(x_1,\ldots,x_n)\in\{0,1\}^{n}: \,\, \sum_{k=1}^{n}x_k = m \}. $$ So, $B_{n, m}$ is the set of $n$-tuples of 0 and 1 with the fixed sum equal to $m$. ($m>1$ is the fixed natural number) $\endgroup$ – monica_k Jun 12 '18 at 8:59

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