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\begin{document}
\title{\vspace{-2em}Geometry of Numbers TCC 2024 Exercises}
\date{\vspace{-2em}\today}
\maketitle
\subsection*{Motivation}
The motivation for these questions is explained in the \href{https://urss.warwick.ac.uk/items/show/251}{summer project of Kate Thomas}.
We are going to study, for a variable $t \in (0,1)$ and a parameter $N>1$ which can be thought of as very large, the quantity
\begin{multline*}
I(t,N) =
\int_{\substack{\alpha\in \Mat^{\Sym}_d(\R)\\ \|\alpha\|<1}} \#\{A,B \in \Mat_{d\times d}(\Z) : \|A\| < N, \|t A \alpha-B\| <1/N\} \\
=
\sum_{A,B \in \Mat_{d\times d}(\Z) : \|A\| < N} \measure\{\alpha\in \Mat^{\Sym}_d(\R): \|\alpha\|<1 ,\|tA \alpha-B\| <1/N\},
\end{multline*}
but we'll work up to it by steps.
\subsection*{Notation}
For an $m \times n$ real matrix $M$, we define the 2-norm $\|M\| = \sqrt{\sum M_{ij}^2}$.
Recall the Smith normal form of $A$,
\[
A = U^{-1} \diag(e_1,\ldots,e_d) V^{-1},
\]
where $U, V \in \SL_n(\Z)$ and $e_i\in \N$ with $e_1\mid \dotsb \mid e_d$.
We will use \href{https://en.wikipedia.org/wiki/Big_O_notation#History_(Bachmann%E2%80%93Landau,_Hardy,_and_Vinogradov_notations)}{big-O/little-o and Vinogradov $\ll$ notation}. You may want to use the ``divisor bound”
\[
\{d\in \N: d|m\} \ll_\eps m^\eps (m \in \N).
\]
In general, in these questions, when you’re asked for an upper bound it’s always OK for it to be multiplied by $O_\eps((\text{some variable})^\eps)$.
\subsection*{Marking}
Out of 100. You are strongly encouraged to collaborate with other students; if you take the course for credit you must write up your answers separately.
25\% for \href{mailto:simon.rydin-myerson@warwick.ac.uk}{sending me} plausible strategies for two questions by the check-in deadline. (2 pages, clearly expressed, you can use more pages if you want.)
75\% for submitting solutions to at least three questions (25 each, best three count). Many questions are open-ended or hard. \textbf{I will be looking only for a plausible strategy followed through to its logical conclusion, whether or not it successfully answers the question.} You are welcome to check with me if you're not sure. If between you all questions get answers, we should almost have a theorem!
\clearpage
\textbf{Answer three questions. (Check-in: Show strategies for two.)}
\begin{enumerate}
\item As a warm-up we’ll count invertible matrices $A$ with $\|A\|1$. Show that the number of an $n\times n$ invertible matrices $A$, with $\|A\|1$, the number of an $n\times n$ invertible matrices $A$, with $\|A\|