%% Use A4 page size, bigger fonts
\documentclass[a4paper,12pt]{amsart}

%% Use also various symbols from the AMS
\usepackage{amsmath,amssymb}

%% A package with additional math symbols
\usepackage{mathtools}

%% A command to typeset the := symbol in a nicer way; uses the
%% mathtools package
\newcommand{\df}{\coloneqq}

%% Decrease the size of the margins to use more of the page
\usepackage{fullpage}

%% The colorlinks option gives colored links instead of boxes around
%% links for equation numbers and other references
%% This can be a problem for color-blind readers
\usepackage[colorlinks]{hyperref}

%% The usual theorems, etc
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem*{theorem*}{Theorem}
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}

%% Style for remarks
\theoremstyle{remark}
\newtheorem{remark}{Remark}

%% Style for definitions
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}

%% Define a new command to typeset GL as a function
\DeclareMathOperator{\gl}{GL}

%% An operator where the subscripts/superscripts appear under/over the
%% symbol
\DeclareMathOperator*{\expect}{\mathbf{E}}

%% For comparison when we use the command without *
\DeclareMathOperator{\expecta}{\mathbf{E}}

%% Allow to put some text over an arrow: this is a command with one
%% argument
\newcommand{\fleche}[1]{\stackrel{{#1}}{\longrightarrow}}

%% A version where both the top and bottom can be specified
\newcommand{\fleches}[2]{\stackrel{{#1}}{{#2}}}

%% Use the AMS versions of the "less than or equal to" and "larger
%% than or equal to" signs
\renewcommand{\leq}{\leqslant}
\renewcommand{\geq}{\geqslant}

%% A command "boldmath" to type bold mathematical formulas in such a
%% way that small letters remain slanted instead of being straight
\newcommand{\bm}[1]{\text{\boldmath${#1}$}}

%% The basic number sets in the bold style
% \newcommand{\C}{\mathbf{C}}
% \newcommand{\R}{\mathbf{R}}
% \newcommand{\Q}{\mathbf{Q}}
% \newcommand{\Z}{\mathbf{Z}}

%% The basic number sets in the "blackboard" style. Using this or the
%% previous is a question of taste, but they shouldn't be mixed.
\newcommand{\C}{\mathbb{C}}
\newcommand{\R}{\mathbb{R}}
\newcommand{\Q}{\mathbb{Q}}
\newcommand{\Z}{\mathbb{Z}}

%% Begin the text
\begin{document}

%% Title
\title{Fourth file}

%% Author
\author{E. Kowalski}
%% Physical address
\address{ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland}
%% Email address
\email{kowalski@math.ethz.ch}

%% Additional authors can be added by repeating the above
% \author{P. Ménard}
% \address{Institute for Borgesian Studies}
% \email{menard@ibs.ow}


\begin{abstract}
  More basic LaTeX commands and guidelines, including bibliography
  without BiBTeX.
\end{abstract}

%% Print the title page

\maketitle

\section{More general LaTeX things}

Binomial coefficients:
$$
\binom{a}{b}=\binom{a}{b-a}.
$$

Ceiling and floor functions, including bigger versions:
$$
\lceil n^{1/2}\rceil,\quad\quad
\Bigl\lfloor \frac{n^{1/2}}{e+1}\Bigr\rfloor.
$$

Overline and underline, conjugate:
$$
\overline{z+w}=\bar{z}+\bar{w},\quad\quad
\underline{a+b}.
$$

Limits:
$$
\lim_{n\to+\infty}\sum_{k=1}^n\frac{1}{n^2}=\frac{\pi^2}{6}.
$$

Force typesetting in ``display'' mode: compare
$$
\frac{\lim_{n\to+\infty}\sum_{k=1}^n\frac{1}{n^2}}
{\lim_{n\to+\infty}\sum_{k=1}^n\frac{1}{n^4}}=15
$$
and
$$
\frac{\displaystyle{\lim_{n\to+\infty}\sum_{k=1}^n\frac{1}{n^2}}}
{\displaystyle{\lim_{n\to+\infty}\sum_{k=1}^n\frac{1}{n^4}}}=15.
$$

Angled brackets with better spacing than just typing $<$ and $>$: compare
$$
<f,g>=\int_0^{+\infty}f(x)\overline{g(x)}dx
$$
with
$$
\langle f,g\rangle=\int_0^{+\infty}f(x)\overline{g(x)}dx.
$$

Defining sets:
%% the \, gives a little extra horizontal space
$$
X=\{x\in\R\,\mid\, |x-\pi|<1\},\quad\quad
X=\{x\in\R\mid |x-\pi|<1\},
$$
or with bigger curly brackets:
$$
X=\Bigl\{x\in\R\,\mid\, \Bigl|\frac{a}{p}-x\Bigr|<1\Bigr\},
$$

Direct sums, tensor product, and the big variants:
$$
E=F\oplus G,\quad\quad E=F\otimes G,\quad\quad
E=\bigoplus_{i\in I}E_i,\quad\quad
E=\bigotimes_{i\in I}E_i.
$$

Using a new operator, with subscripts/superscripts under/over:
$$
\expect_{x\in X}f(x),\quad\text{compared with}\quad \expecta_{x\in X}f(x).
$$

Stacking information:
$$
X\fleche{f} Y,\quad E\fleche{a+b} F,\quad\quad
X\fleches{f}{\longleftarrow} Y.
$$

More symbols:
$$
H\rtimes G,\quad\quad H\ltimes G,\quad\quad H\triangleleft G,\quad\quad
H\triangleright G.
$$

Norm:
$$
\| x\|,\quad  \text{ instead of } \quad ||x||.
$$

\section{Bibliography, without Bibtex}

K. Matomäki has obtained the best known results concerning the
distribution of $\{\alpha p\}$, see \cite{matomaki}.

The work of Perelman \cite{perelman} has revolutionized
$3$-dimensional geometry and topology.

Vinogradov, in his book~\cite{vinogradov}, obtained the first
asymptotic formulas for oscillating sums over primes. For instance
%% The square brackets are used to add additional information,
%% typically a location in the text cited
(see \cite[Ch.\,XI]{vinogradov}) he proved that
$$
\sum_{\{xp\}<c}\frac{1}{p}=+\infty
$$
for any irrational number $x$.

%% The Vin indicates the widest a bibliographical reference may be,
%% for horizontal spacing of the bibliography.
\begin{thebibliography}{Vin}

  %% A research paper.  It will appear with just a number as
  %% reference, which will be by \cite{matomaki}
\bibitem{matomaki} K. Matomäki: \textit{The distribution of $\alpha p$
    modulo one}, Math. Proc. Cambridge Philos. Soc. 147 (2009),
  267--283.

  %% A preprint on arXiv.
\bibitem{perelman} G. Perelman: \textit{The entropy formula for the
    Ricci flow and its geometric applications}, preprint (2002),
  \url{https://arxiv.org/abs/math/0211159}
  
  %% A book; the letters in square brackets will appear as the identifier
  %% in citations.
\bibitem[Vin]{vinogradov} I.M. Vinogradov: \textit{The method of
    trigonometrical sums in the theory of numbers}, Interscience
  Publishers, 1963.

\end{thebibliography}

\end{document}
