\documentclass[12pt]{amsart}

%================DIMENSIONS======================
%\oddsidemargin 0in 
%\evensidemargin 0in
%\textwidth 6.5in
%\topmargin 0in
%\headheight .21in
%\headsep .29in   % \headheight + \headsep = 0.5in
%\textheight 8in % \textheight includes \footskip
%\footskip .5in 
%\skip\footins 22.8pt plus 4pt minus 2pt
%\parskip 5 pt plus 1.5pt minus .5pt
%\itemsep 0 pt plus .5pt
%================DIMENSIONS======================

\usepackage{amsmath,amsthm,amsfonts,amscd,amssymb}

\newtheorem{thm}{Theorem}[section]
\newtheorem{cor}[thm]{Corollary}
\newtheorem{lem}[thm]{Lemma}
\newtheorem{prop}[thm]{Proposition}
\newtheorem{ax}{Axiom}
\newtheorem{prob}{Problem}
\newtheorem{quest}{Question}

\theoremstyle{definition}
\newtheorem{defn}{Definition}[section]

\theoremstyle{remark}
\newtheorem{rem}{Remark}[section]
\newtheorem*{notation}{Notation}

\begin{document}
\title{MATH 195, Spring 2019, Take-home midterm}

\maketitle


\def\A{{\mathcal A}}
\def\B{{\mathcal B}}
\def\C{{\mathcal C}}
\def\D{{\mathcal D}}
\def\F{{\mathcal F}}
\def\x{{\mathcal H}}
\def\I{{\mathcal I}}
\def\J{{\mathcal J}}
\def\K{{\mathcal K}}
\def\L{{\mathcal L}}
\def\M{{\mathcal M}}
\def\R{{\mathcal R}}
\def\s{{\mathcal S}}
\def\V{{\mathcal V}}
\def\X{{\mathcal X}}
\def\Y{{\mathcal Y}}
\def\H{{\mathcal H}}
\def\OO{{\mathcal O}}
\def\BB{{\mathbb B}}
\def\cee{{\mathbb C}}
\def\pee{{\mathbb P}}
\def\que{{\mathbb Q}}
\def\real{{\mathbb R}}
\def\zed{{\mathbb Z}}
\def\hyp{{\mathbb H}}
\def\qbar{{\overline{\mathbb Q}}}
\def\eps{{\varepsilon}}
\def\ahat{{\hat \alpha}}
\def\bhat{{\hat \beta}}
\def\gt{{\tilde \gamma}}
\def\h{{\tfrac12}}
\def\be{{\boldsymbol e}}
\def\bei{{\boldsymbol e_i}}
\def\baf{{\boldsymbol f}}
\def\ba{{\boldsymbol a}}
\def\bc{{\boldsymbol c}}
\def\bm{{\boldsymbol m}}
\def\bk{{\boldsymbol k}}
\def\bi{{\boldsymbol i}}
\def\bj{{\boldsymbol j}}
\def\bl{{\boldsymbol l}}
\def\bq{{\boldsymbol q}}
\def\bu{{\boldsymbol u}}
\def\bt{{\boldsymbol t}}
\def\bs{{\boldsymbol s}}
\def\bv{{\boldsymbol v}}
\def\br{{\boldsymbol r}}
\def\bw{{\boldsymbol w}}
\def\bx{{\boldsymbol x}}
\def\bX{{\boldsymbol X}}
\def\bz{{\boldsymbol z}}
\def\bwy{{\boldsymbol y}}
\def\bY{{\boldsymbol Y}}
\def\bL{{\boldsymbol L}}
\def\bb{{\boldsymbol b}}
\def\bet{{\boldsymbol\eta}}
\def\bxi{{\boldsymbol\xi}}
\def\bo{{\boldsymbol 0}}
\def\bol{{\boldkey 1}_L}
\def\ep{\varepsilon}
\def\p{\boldsymbol\varphi}
\def\q{\boldsymbol\psi}
\def\rank{\operatorname{rank}}
\def\aut{\operatorname{Aut}}
\def\lcm{\operatorname{lcm}}
\def\sgn{\operatorname{sgn}}
\def\spn{\operatorname{span}}
\def\md{\operatorname{mod}}
\def\Norm{\operatorname{Norm}}
\def\dim{\operatorname{dim}}
\def\det{\operatorname{det}}
\def\Vol{\operatorname{Vol}}
\def\rk{\operatorname{rk}}
\def\GL{\operatorname{GL}}


\begin{center}
{\bf Please print your name clearly!}
\end{center}
\bigskip

\noindent
{\bf Name:\underline{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \  \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }}
\bigskip

\noindent
{\it This test is due on Thursday, 3/28/19, in class. While completing it, feel free to use your lecture notes from class, as well as those posted on the class webpage. You are however not allowed to consult with anyone: it is understood that solutions to this midterm represent solely your own work with no outside assistance. Good luck!}
\bigskip


\noindent
{\bf Problem 1. (10 points)} Let $\Lambda$ be a lattice of rank $r$ in $\real^n$, $1 \leq r < n$, and let $V = \spn_{\real} \Lambda$. Prove that $\Lambda$ is a discrete co-compact subgroup~of~$V$. 

\newpage



\noindent
{\bf Problem 2. (15 points)} Let $\Lambda$ be a lattice of full rank in~$\real^n$, $n \geq 1$. Let $M$ be a compact convex $\bo$-symmetric set in~$\real^n$ with $\Vol(M) > 0$. Define a counting function
$$f(t) = \left| tM \cap \Lambda \right|,$$
i.e. $f(t)$ gives the number of points of $\Lambda$ in the homogeneous expansion of $M$ by $t$. Prove that
$$\lim_{t \to \infty} \frac{f(t)}{t^n} = \frac{\Vol(M)}{\det (\Lambda)}.$$

\newpage


\noindent
{\bf Problem 3. (15 points)} Let $a,b$ be positive real numbers, and suppose that 
$$\Lambda = \left( \begin{matrix} a&b&0&0 \\ 0&1&a&b \\ 0&0&1&1 \\ 1&0&0&1 \end{matrix} \right) \zed^4$$
is a full-rank sublattice of $\zed^4$.
\smallskip

\noindent
{\bf a) - (5 points)} What are all the possible values of $a$ and $b$? Prove your answer.
\smallskip

\noindent
{\bf b) - (5 points)} Suppose that $a=b$ and $\Omega$ is a full-rank sublattice of $\Lambda$, such that the volume of a fundamental domain of $\Omega$ in $\real^4$ is equal to 20. What are all the possible values of $a$? Prove your answer.
\smallskip

\noindent
{\bf c) - (5 points)} Assuming part b, can $\Lambda$ be a sublattice of any of the following two lattices:
$$L_1 = \left( \begin{matrix} 1&0&0&0 \\ 0&1&0&1 \\ 0&0&1&1 \\ 1&0&0&2 \end{matrix} \right) \zed^4,\  L_2 = \left( \begin{matrix} 1&0&0&0 \\ 0&1&0&1 \\ 0&0&1&1 \\ 1&0&0&3 \end{matrix} \right) \zed^4,$$
and if so, which one(s)? What are all the possible values of $a$ in each case? Prove your answers.

\newpage



\noindent
{\bf Problem 4. (10 points)} Let $n \geq 2$ be even, and define quadratic forms
$$Q_1(X_1,\dots, X_n) = \sum_{i=1}^n X_i^2 - X_1X_2 - X_3X_4 - \dots - X_{n-1}X_n,$$
and
$$Q_2(X_1,\dots, X_n) = \bX^t B \bX$$
for some real symmetric matrix $B$. Suppose that $Q_1$ and $Q_2$ are isometric. 
\smallskip


\noindent
{\bf a) - (5 points)} Find eigenvalues of $B$. Prove your answer.
\smallskip

\noindent
{\bf b) - (5 points)} Let
$$Q'_1(X_1,\dots, X_n) = \frac{1}{2} \sum_{i=1}^n X_i^2 - X_1X_2 - X_3X_4 - \dots - X_{n-1}X_n.$$
Are $Q'_1$ and $Q_2$ isometric? Prove your answer. 




\newpage
\noindent
{\bf Problem 5. (10 points)} Let $n \geq 2$, and let $F : \real^n \to \real$ be a continuous function such that

\begin{enumerate}

\item $F(\bx) \geq 0$ for all $\bx \in \real^n$,

\item $F(a \bx) = |a| F(\bx)$ for all $a \in \real$ and $\bx \in \real^n$.

\end{enumerate}
Let
$$X = \{ \bx \in \real^n : F(\bx) \leq 1 \}.$$
Assume additionally that $F$ satisfies the triangle inequality:
\begin{equation}
\label{tr}
F(\bx+\bwy) \leq F(\bx) + F(\bwy),
\end{equation}
for all $\bx,\bwy \in \real^n$. Let $\Lambda$ be a lattice of full rank in $\real^n$. Prove that for every real number
$$\mu \geq 2 \left( \frac{\det(\Lambda)}{\Vol(X)} \right)^{1/n}$$
the intersection $\mu X \cap \Lambda$ contains a non-zero vector. Is this statement still true if $F$ does not satisfy the triangle inequality~\eqref{tr}? Either prove your answer or give a counter-example.




\newpage
\noindent
{\bf Problem 6. (10 points)} Prove that the hexagonal lattice is perfect and eutactic. Further, prove that a lattice $L$ in $\real^2$ is perfect if and only if it is similar to the hexagonal lattice.


\newpage

\noindent
{\bf Problem 7. (10 points)} Let $\Lambda$ be a lattice of full rank in $\real^n$, and let $M \subset \real^n$ be a compact convex set such that $\Vol(M) < \det(\Lambda)$. Prove that there exists a point $\bx \in \real^n$ such that the intersection $M \cap (\Lambda+\bx)$ is empty.


\newpage


\noindent
{\bf Problem 8. (10 points)} Let $\{n_j\}_{j=1}^{\infty}$ be a sequence of natural numbers such that
$$\lim_{j \to \infty} \frac{n_{j+1}}{n_j} = \infty.$$
Define
$$f(x) = \sum_{j=1}^{\infty} x^{n_j},$$
and let $\alpha$ be a real algebraic number such that $0 < \alpha < 1$. Prove that $f(\alpha)$ is transcendental.



\end{document}