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main.tex
8
main.tex
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For errata and more stuff, see \url{https://sarielhp.org/book/}
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\section{Grid}
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\begin{oneshot}{1.1}
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\begin{exercise}\label{ex1.1}
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Let $P$ be a max cardinality point set contained in the $d$-dimensional unit hypercube such that the smallest distance of point pairs in $P$ is 1. Prove that
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\[\left( \floor{\sqrt{d}}+1 \right)^d \leq |P|\leq \left( \ceil{\sqrt{d}}+1 \right)^d. \]
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\end{oneshot}
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\end{exercise}
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hmm... the first exercise in this book is wrong. See \url{https://sarielhp.org/book/errata.pdf}.
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The stated lowerbound is actually an upperbound.
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\begin{proof}
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@@ -36,9 +36,9 @@ n &\geq 1/\vol(1b^d)\\
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The last line is greater than $(\sqrt{d}/5)^d$ for large enough $d$.
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\end{proof}
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\begin{oneshot}{1.2}
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\begin{exercise}
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Compute clustering radius
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\end{oneshot}
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\end{exercise}
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\end{document}
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