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8
main.tex
8
main.tex
@@ -15,10 +15,10 @@
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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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@@ -26,12 +26,14 @@
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\setCJKsansfont{Source Han Serif SC}
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\setCJKmonofont{Source Han Serif SC}
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\RequirePackage[charter]{mathdesign} % math font only for serif mode
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\RequirePackage{amsmath}
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\RequirePackage{eucal}
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\RequirePackage{microtype}
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\fi
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% \setCJKmainfont{SourceHanSerifSC-Regular.otf}[BoldFont={SourceHanSerifSC-Bold.otf}, ItalicFont=SourceHanSerifSC-Regular.otf]
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\RequirePackage[margin=1in]{geometry}
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\RequirePackage{amsmath}
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\RequirePackage[dvipsnames]{xcolor}
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\RequirePackage[colorlinks=true,urlcolor=Blue,citecolor=Green,linkcolor=BrickRed,unicode]{hyperref}
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@@ -43,8 +45,6 @@
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\RequirePackage{graphicx} % essentials
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% \RequirePackage{microtype}
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\RequirePackage{eucal}
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\RequirePackage[nocompress]{cite} % other convenient stuff
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\RequirePackage{enumerate}
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%\usepackage{pgf,tikz}
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@@ -57,18 +57,21 @@
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\newaliascnt{corollary}{theorem}
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\newaliascnt{conjecture}{theorem}
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\newaliascnt{proposition}{theorem}
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\newaliascnt{exercise}{theorem}
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\newtheorem{lemma}{Lemma}[section]
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\newtheorem{corollary}{Corollary}[section]
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\newtheorem{conjecture}{Conjecture}[section]
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\newtheorem{proposition}{Proposition}[section]
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\newtheorem{problem}{Problem}
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\newtheorem{exercise}{Ex}[section]
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\newcommand{\lemmaautorefname}{Lemma}
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\newcommand{\corollaryautorefname}{Corollary}
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\newcommand{\conjectureautorefname}{Conjecture}
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\newcommand{\propositionautorefname}{Proposition}
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\newcommand{\problemautorefname}{Problem}
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\newcommand{\exerciseautorefname}{Exercise}
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\newenvironment{definition}[1][Definition]{\begin{trivlist}
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\item[\hskip \labelsep {\bfseries #1}]}{\end{trivlist}}
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