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\begin{document}

%% Title
\title{Some simple Stylistic guidelines}


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

\begin{abstract}
  A summary of some stylistic guidelines.
\end{abstract}

\maketitle

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\section{Introduction}

Mathematical texts exist for different \emph{purposes}, and different
\emph{audiences}. Before starting to write, it is important to think
about both of these aspects -- which are related.

%% Some slightly larger vertical space as a single paragraph change.
\medskip

The main possible purposes are the following:
\begin{itemize}
\item Working out some mathematical problem for yourself.
\item Sketching some argument to show to colleagues.
\item Preparing the content of a lecture, whether a course or a single
  talk.
\item Writing about \emph{known} results to explain them.
\item Writing \emph{new} results.
\end{itemize}

\medskip

The target audience could be the following (and is often a combination
of some of them):
\begin{itemize}
\item Yourself only.
\item Some colleagues.
\item Students at various levels.
\item Experts in a certain area of mathematics.
\item Professional mathematicians working in related areas.
\item Professional mathematicians working in different fields.
\end{itemize}

\medskip

The form in which the text is presented will often also be related to
these two aspects:
\begin{itemize}
\item Private notes.
\item Letters or emails.
\item Research paper (the most frequent form of \emph{published}
  mathematics).
\item Survey paper, including most bachelor or master theses.
\item Textbook for students.
\item Textbook for professional mathematicians.
\item Research monographs.
\end{itemize}

\begin{remark*}
  (1) Mathematics is also communicated in talks (discussions between a few
  mathematicians, or lectures at all levels).  How to do this well is
  another important question.

  (2) \textbf{All the items below are just \emph{guidelines}. Each may sometimes
  reasonably be ignored, and each is fairly often ignored in actual
  mathematical texts (including very famous ones).}
\end{remark*}

\subsection*{Acknowledgments} I thank the students of the HS 2023
``Mathematical writing'' lecture at ETH for their interest and feedback.

\section{General guidelines}

\begin{enumerate}
\item A mathematical text is in particular a text, and you should think
  of it partly this way, and try to follow general guidelines for
  ``good'' writing (grammar, spelling, sentence length, repetitions,
  etc). It can be very useful to
  \begin{enumerate}
  \item Read parts of the text aloud;
  \item Show it to some friends or colleagues and ask for feedback;
  \item Leave the text as it is for a while and re-read it later without
    hurry; this can help detect material that seemed perfectly clear at
    one point, but is in fact confusing or simply wrong;
  \item Run the spell-checker, at least at the end of the writing.
  \end{enumerate}

\item As in any text, simple typographical or spelling mistakes occur
  frequently. Re-reading also serves the purpose of finding such
  problems.

\item As in literature, some authors are ``better'' than others,
  independently of the mathematical content, and one can learn by
  reading and paying attention to ``good'' writers (one can also learn
  from bad writers).  The citations statistics of Mathscinet and zBmath
  can be useful: if a book has been cited in hundreds or thousands of
  papers, this usually means that it is rather well-written.
  
\item Avoid the use of words like ``trivial'', ``obvious'', ``clear'';
  in fact, mistakes often hide exactly in such places.

  When pointing out that some fact or argument is, indeed, very easy, an
  option is to state something like: ``By definition, we have...'', or
  ``By construction, ...''. This indicates to the reader that what
  follows \emph{should} be straightforward.

\item Avoid very vague justifications, which might be fairly clear to
  experts in one area, but are especially confusing to a student, for
  instance, ``By a compactness argument, there exists $\delta>0$ such
  that...'', or ``By class-field theory, the cardinality of the set
  $S$ is even''.  It is usually not very difficult to be more precise,
  even if one really doesn't want to spell out completely a routine
  step.

\item When making statements involving multiple parameters or data,
  it is usually best to introduce them from the top down. The
  quantification should also be clear, and the nature and possible
  restrictions on the data also. Some notation can be explained at the
  end, especially if it is likely to be unambiguous anyway.

  For instance, compare
  \begin{squote}
    Let $n\geq 1$ and $m\geq 1$ be integers. Let $U\subset \R^n$ be an
    open subset. Let $f\colon U\to \R^m$ be a continuous function. For
    any compact subset $K$ of~$U$ and for any real number $\eps>0$,
    there exists $\delta>0$ such that for $x$ and $y$ in~$K$, the
    condition $\|x-y\|<\delta$ implies $\|f(x)-f(y)\|<\eps$, where
    $\|a\|$ denotes the euclidean norm of a vector.
  \end{squote}
  with
  \begin{squote}
    If $K\subset U\subset \R^n$ is compact and $f\colon U\to \R^m$ is
    continuous then $f(K)$ is compact.
  \end{squote}
  
\item In algebra in particular, avoid words like ``canonical morphism'',
  and make sure that it is always clear what kind of map or morphism is
  meant (for instance, is a map $f\colon \R^n\to \R^m$ supposed to be a
  linear map? a continuous map? a ring morphism?).

  Be careful with isomorphisms used too loosely:

  \begin{example}
    We have $\Z\simeq 2\Z$, so $\Z/\Z\simeq \Z/2\Z$, so $1=2$.
  \end{example}

\item In mathematics, asking \emph{good questions} is often as important
  as answering them. In particular, papers which open new directions by
  suggesting new problems are often more interesting than those which
  seem to completely close a subject.
 
\end{enumerate}


\section{Minor typographical guidelines}

\begin{enumerate}
\item Do not state a theorem which consists entirely of a displayed
  formula.  Add at least few words at the beginning (which often should
  be done anyway to state assumptions).


  \begin{theorem}
    
    $$\ell(D)-\ell(K-D)=\deg(D)+1-g.$$
  \end{theorem}
  
\item Do not start a sentence with a mathematical formula or symbol.

  \begin{squote}
    $u$ is linear map on $E$. $u$ therefore is injective if and only if
    its kernel is zero.
  \end{squote}

\item Always put mathematical content between dollar signs, even if it
  is a single letter, since the typography is not the same.

  \begin{squote}
    Let n be an integer. Let $m$ be another integer.
  \end{squote}

\item Spell out numbers when they are used in ``ordinary sentences'':
  
    \begin{squote}
    We present two different proofs of this theorem. They involve $2$
    different techniques.
  \end{squote}
  
\item Do not forget fullstops at the end of formulas:
  $$
  \ell(D)-\ell(K-D)=\deg(D)+1-g.
  $$

\item Use quotes in the right directions: ``right'' (typed
  \verb|``right''|) instead of "wrong" (typed \verb|"wrong"|).

\item The tilde or hat  signs on top of letters produced by
  \verb|\tilde| or \verb|\hat| is often small and hard to read. The wide
  variants can then we use; they also extend over multiple characters,
  which can be very useful.

  $$
  \tilde{x},\quad \hat{f},\quad \hat{f*g},\quad
  \widetilde{x},\quad \widehat{f},\quad \widehat{f*g}.
  $$
  
\item Use em-dashes \verb|--| (two minus signs) instead of single
  \verb|-| (one minus sign) to separate names or page numbers.

  \begin{example}
    Write ``p. 230--234'' instead of ``p. 230-234'', or ``the
    Gromov--Hausdorff distance'' instead of ``the Gromov-Hausdorff
    distance''.
  \end{example}
  
\item If you are going to try to publish your text, it is best not to
  have too much typographical sophistication (with many LaTeX packages,
  etc) since these might not be compatible with the publisher's
  setup. This includes also preprint servers such as
  \href{https://arxiv.org}{arXiv} where you have to submit the LaTeX
  file.

\item Pay attention to people's names in the bibliography. If they
  include accented letters, for instance, make sure to include them. If
  you are not sure of the preferred spelling (for names transliterated
  from a different writing system or alphabet), copy what is done in one
  of their works or in some other place where the name is quoted.

  Compare Erd\H os and Erdös.
\end{enumerate}


\section{Notation and terminology}

\begin{enumerate}
\item You should choose notation and terminology carefully, and think
  about it.
  
\item In particular, when objects depend on a lot of data previously
  introduced, think carefully about how to present this dependency:
  should it be explicit (as argument of a function, or subscript, or
  superscript), or left implicit?

  \begin{squote}
    Let $X$ be a finite set of real numbers. Let $d\geq 1$ and $k\geq 1$
    be integers. For each element $x\in X$, let $n_x$ be a non-negative
    integer such that $0\leq n_x\leq d$. Let $\mathcal{F}$ be the space
    of functions of class $C^k$ such that $f^{(j)}(x)=0$ for all $x\in
    X$ and all $j$ such that $0\leq j\leq n_x$.
  \end{squote}

  compared with

  \begin{squote}
    Let $X$ be a finite set of real numbers. Let $d\geq 1$ and $k\geq 1$
    be integers. For each element $x\in X$, let $n_x$ be a non-negative
    integer such that $0\leq n_x\leq d$. Let $\mathcal{F}(X,d,k,n)$ be
    the space of functions of class $C^k$ such that $f^{(j)}(x)=0$ for
    all $x\in X$ and all $j$ such that $0\leq j\leq n_x$.
  \end{squote}

  If what follows is a statement where the most important fact is that
  $k$ is a parameter tending to infinity (a property of the spaces
  $\mathcal{F}$ as $k\to +\infty$ with the other parameters unchanged),
  then it would be best to use a notation like $\mathcal{F}_k$.

\item When taking over material from different sources, you should
  ensure that the notation and terminology are consistent and uniform.

\item For key concepts, it is useful to use specific macros which convey
  some semantic meaning in the LaTeX file, and whose typographical
  representation can also be changed later:

  \begin{squote}
\begin{verbatim}
We will write $\gowers{n}{f}$ for the $n$-th Gowers
norm of a function $f$.
\end{verbatim}
  \end{squote}


  LaTeX already knows many such commands, and there are many other extra
  packages with more symbols, commands, etc.
  
\item In choosing notation, be careful to avoid clashes, like using the
  same letter for two different things (for instance, avoid having $p$
  denote both a prime number and an exponent in an $L^p$-space).

\item Be careful about which objects you will denote with capital
  letters, and which ones will have lowercase letters. This affects the
  esthetic, but also readability (including sometimes mathematical
  readability).

  \begin{example}
    Compare
    $$
    |a|\leq CK^C|b| \quad \text{ with } \quad
    |A|\leq c\alpha^d|B|.
    $$
  \end{example}
  
\item You can use different font styles to distinguish different
  classes of objects; if done consistently, this is very helpful.

  \begin{example} In integration theory, you could use a convention that
    curly letters refer to sets of subsets ($\sigma$-algebras, etc),
    capital letters to subsets, and lowercase letters to elements:
  \begin{squote}
    Here $\mathscr{F}$ is a $\sigma$-algebra, $E\in\mathscr{F}$ is a
    measurable set, and $x\in E$ an element of $E$.
  \end{squote}

  (Note that even then, adding some words to always indicate what class
  of object is involved is useful.)

  Compare with:
  \begin{squote}
    Let $x$ be a $\sigma$-algebra, $\mathcal{F}$ a set in
    $x$ and $E$ is element of $\mathcal{F}$.
  \end{squote}
\end{example}

\item Do not enumerate a finite or countable set, unless or until you
  need to.

  \begin{example}
    A sum over primes $p$ up to a bound $x$ is better written
    $$
    \sum_{p\leq x}f(p)
    $$
    instead of introducing the notation $p_n$  for the $n$-th prime and
    writing
    $$
    \sum_{n=1}^kf(p_n).
    $$

    A basis of a finite-dimensional vector space can be denoted
    $(e_i)_{i\in I}$ instead of $(e_1,\ldots, e_n)$.
  \end{example}

\item If you introduce some terminology, try to make it
  \emph{searchable} and natural (in particular, neither too generic nor
  too strange). This should for instance help readers find the
  definition in the text, and also makes it possible to change your mind
  if needed.

  \begin{example}
    Do not split a set in two subsets by saying that some of the
    elements are ``good'' and the others ``bad''. (It is too generic,
    among other problems.)
  \end{example}
  
\item Bad terminology does exist, and is sometimes unavoidable, but try
  to use such terminology sparingly. Outside of research papers or
  books, it can be fine to introduce a different convention than what is
  ``standard''.

  \begin{example}
    A polymatroid is not a special case or generalization of a matroid,
    but is a polytope associated to a matroid.
  \end{example}

\item Try to be aware of the existence of slightly different conventions
  for certain definitions, and be ready to clarify which one you use.

  \begin{example}
    In the French mathematical literature (and in much of what French
    authors write in English), ``compact'' means ``compact and
    Hausdorff'' (and ``a Hausdorff space'' is most often ``a separated
    space''). Also, in French texts, ``positive'' means
    ``non-negative'', and ``strictly positive'' is used for $>0$.
  \end{example}

  \begin{example} The meaning of $\N$ is not absolutely standardized
    (is~$0$ there or not?). The notation $\N_0$ is unknown in France or
    most of the US. A good way to avoid ambiguity is to uses
    ``self-explanatory'' notation like $\Z_{\geq 1}$ for positive
    integers and $\Z_{\geq 0}$ for non-negative integers.
  \end{example}

  \begin{example}
    Many people use $x\gg 0$ to indicate that $x$ is very large. In
    other communities (e.g., analytic number theory), however, $f\ll g$
    usually means that $|f(x)|\leq Cg(x)$ for some constant $C\geq 0$,
    and $g\gg f$ means that $f\ll g$.
  \end{example}
  
\item In general, do not change any notation or terminology that is
  completely standard, unless you have a very good reason for it. You
  should then \emph{explain} why you are doing the changes.

  \begin{squote}
    Following Bourbaki, we will use the terminology ``stellar algebra''
    for what is usually called a $C^*$-algebra.
  \end{squote}

\end{enumerate}

\section{Explaining}

\begin{enumerate}
\item If you are presenting a result which is \emph{already known}, the
  goals include:
  \begin{enumerate}
  \item Motivate the result, explain what it means.
  \item Explain the ideas of the proof.
  \item Give context and background, including applications.
  \end{enumerate}
  
\item If the result is ``well-known'', making sure that the proof you
  present is complete and rigorous is not absolutely necessary, and may
  obscure the key ideas and insights involved.

\item One common technique in textbooks and survey papers is to make
  simplifying assumptions in the proof.  It is then good practice to
  point it out, and to give a reference to a source where the full proof
  is given.

  \begin{squote}
    In this proof, we assume for simplicity that the group $G$ is a
    countable union of compact sets. The general case is treated, for
    instance, in~\cite[\S\,1.2,\,Th.\,1]{bourbaki} by Bourbaki.
  \end{squote}

  or
  
  \begin{squote}
    The Growth Theorem is easier to state in the case of $\SL_n(\F_p)$,
    and we present this special case first.
  \end{squote}

\item Even very complicated objects can often be defined quite quickly
  using a few successive definitions. But usually, even if each
  individual step is elementary, the final outcome may be
  incomprehensible to most readers.

\item Statements of theorems can also be very complicated and difficult
  to parse and understand, for instance when given in very great
  generality with many parameters.

  Possible solutions:
  \begin{enumerate}
  \item Start by stating a particularly important special
    case which is more accessible. People often speak of the ``simplest
    non-trivial example'' of a problem or result. 
  \item Add self-contained corollaries explaining what the theorem says
    in important special cases.
  \end{enumerate}

\item Other useful things to think about:
  \begin{enumerate}
  \item Adding explanatory examples, presented in some detail. Following
    a sequence of definitions and arguments as they apply to a single
    well-chosen example is a good idea.
  \item Mention applications to other parts of mathematics (or
    sciences). It is best to add references then.
  \end{enumerate}

\item Do not hesitate to include a picture if there is a good
  illustration of the ideas behind an argument (not only in geometry);
  be careful that the picture \emph{really} is enlightening.

\item To present a complicated proof, try to split it in understandable
  substeps. Each of these can be stated as a separate claim, or as a
  separate lemma. The reader can then first check that the combination
  of the claims gives the final result, and then focus of the different
  steps in turn. If doing this, make sure that the assumptions and
  setting at each substep are clear.

  \begin{example}
    ``Our goal is to show that the group $G$ is the symmetric group
    $S_n$.  \smallskip
    
    \textbf{Step 1.} We show that $G$ acts primitively on $n$ letters.
    (\ldots)

    \medskip
    
    \textbf{Step 2.} We now claim that $G$ contains a transposition.
    (\ldots)

    \medskip

    The result now follows by combining Steps 1 and 2 with a result of
    Jordan, according to which \ldots.''

  \end{example}
      
\end{enumerate}

\section{Convincing}

\begin{enumerate}
\item If you prove a new result and write it down, there are
  different/additional issues to consider. You want to:
  \begin{enumerate}
  \item Convince the reader that the proof is \emph{correct}.
  \item Convince the reader that the result is \emph{interesting}.
  \item Convince the reader that there are \emph{new ideas} in the
    proof.
  \item Explain clearly what these new ideas are.
  \end{enumerate}

\item It can be very useful to add an informal section discussing
  briefly the main \emph{new} ideas of the paper. The ``deeper'' the
  result you are claiming is, the more important it is to convince the
  reader that you have indeed a significant new idea. This applies even
  more when it is a solution of a known problem.
  
\item Keep in mind the possibility that your paper and result might be
  used and cited later on by other researchers.  In particular, state
  the main results in a way that is as self-contained as possible.

  Here is a bad example:
  \begin{quote}
    %% Theorem adds extra vertical space, which we don't need here.
    \vspace{-1.5em}
    \begin{theorem}
      Let~$E$ be a finite dimensional normed vector space. With $\delta$ as
      defined in Section~5, there exists a subspace $F$ of $E$ such that
      the $\delta$-rotundity of the unit ball of $E$ is at least
      $\log\dim(E)$.
    \end{theorem}
  \end{quote}

\item Try to present the main statements as quickly as possible -- a
  lengthy introduction is not always a good idea.

\item Make sure that the statements of your main results do not have
  ``trivial'' counterexamples (for instance, what happens if some set is
  empty, or some vector space is $0$, or some function is constant?).

\item Make sure that the quantification of the various data is clear.

\item Many statements have assumptions whose relevance or necessity are
  not immediately transparent. It is then useful:
  \begin{enumerate}
  \item To make sure that the location within the proof where the
    assumptions are \emph{used} is very clear; in particular, if this
    happens through a reference to another paper, indicate this fact
    when citing this other paper.
  \item To comment whether the assumptions are really needed (or seem to
    be), and not just convenient.
  \end{enumerate}

  \begin{example}
    You might state things like:
    \begin{squote}
      Since $X$ is a Kähler
      manifold, a result of Voisin (see [V1996, Th. 3]) implies that ...
    \end{squote}
    for the first point, or
    \begin{squote}
      We don't know if the assumption that $X$ is a manifold is really
      needed for the conclusion.
    \end{squote}
    or
    \begin{squote}
      We expect that the theorem holds under the weaker assumption that
      $X$ is a topological manifold.
    \end{squote}
    for the second
  \end{example}
  
\item Many statements are also not ``best possible''. It is useful to
  indicate when this is so (even if it is not certain).

  \begin{squote}
    We believe that the conclusion should be valid with any $\delta<1$,
    but we do not see how to prove (or disprove) this.
  \end{squote}
  
\item In a technical proof with many parameters, it is often easier to
  \emph{follow} the proof if the values are presented as soon as they
  are introduced (in terms of previous quantities). If is however often
  easier to \emph{understand} a proof if the role of the parameters is
  explained and their choice is done at the end. It can be a delicate
  balance to select one technique instead of another.

\item Sometimes, a proof might depend on computer help. This should be
  explained carefully, and tools should be made available to help
  readers reproduce the computations if they wish.  Moreover, a
  discussion of \emph{why the computations can be trusted} should be
  included (e.g., the result of the computation can sometimes be checked
  independently at the end -- as in factorizing an explicit number --,
  or parts of the computations can be checked by hand or using
  independent implementations, etc).

  (On the other hand, when computers are used for illustration purposes,
  it is useful to give as much information as possible, but \emph{not
    necessary} to have source code, etc, available.)

\item Data sets or programs or algorithms which are used in computations
  should be part of the bibliography, and reliable long-term
  accessibility should be ensured (institutional repository, GitHub,
  etc).
  
\end{enumerate}

\section{Bibliographic guidelines}


\begin{enumerate}
\item Any important mathematical input must be ackowledged. If this
  input is not in the form of writing documents (or web pages), but
  consists of personal discussions or other informal inputs, then it
  should be mentioned in some acknowledgment.
  
\item It is recommended to give general background references, even when
  not directly necessary to reading and checking a specific result. 
  
\item Citing a paper in general:
  \begin{squote}
    K. Matomäki has obtained the best known results concerning the
    distribution of $\{\alpha p\}$, see \cite{matomaki}.
  \end{squote}

  Here, a preprint:
  \begin{squote}
    The work of Perelman \cite{perelman} has revolutionized
    $3$-dimensional geometry and topology.
  \end{squote}

  And here a book, with indication of where to find a result:
  \begin{squote}
    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 $x$.
  \end{squote}
  
\item When citing a paper or book, it is usually good practice to
  mention the names of the authors.  This is especially true if you are
  using a bibliography style which only prints numbers for the
  citations:

  \begin{squote}
    In~\cite{polya}, it is proved that the simple random walk
    in two dimensions is recurrent.
  \end{squote}

  Compare with
  \begin{squote}
    Pólya (see~\cite{polya}) proved that the $2$-dimensional simple random
    walk is recurrent.
  \end{squote}

\item Any formal statement that is used in a proof should come with
  precise location information. This can be relaxed a bit for "standard"
  facts.
  \begin{squote}
    Deligne's Fourier transform preserves weights
    (see~\cite[Th.\,2.2.1]{katz-laumon}), and therefore \ldots
  \end{squote}
  
\item 
  For results which are usually well-known in a certain field, the
   citation may not be to the original place where it was proved. It is
   good to indicate this.
   \begin{squote}
     Since the congruence subgroup $\Gamma(2)$ of $\SL_2(\Z)$ is free (see,
     e.g.,~\cite[Example\,II.25]{de-la-harpe}), we deduce that \ldots
   \end{squote}
   
 \item Standard facts in the area can be used without references.
  \begin{squote}
    By the dominated convergence theorem, we get $\int f(x)dx=0$.
  \end{squote}

\item Technical note: if you use BiBTeX, always check after compiling if
  all the information you want to display appears!  Also, check capital
  letters and accentuated letters, since BiBTeX can mess them up.
  
\end{enumerate}

\begin{thebibliography}{CC}

\bibitem{bourbaki} N. Bourbaki: \textit{Éléments de mathématique,
    Intégration, chapitre VII}, Springer, 2012.
    
\bibitem{de-la-harpe} P. de la Harpe: \textit{Topics in geometric group
    theory}, Chicago Lectures in Mathematics, University of Chicago
  Press, 2000. 

\bibitem{katz-laumon} N.M. Katz and G. Laumon: \textit{Transformation de
    {F}ourier et majoration de sommes exponentielles}, Publ. Math. IHÉS
  62 (1985), 361--418.

\bibitem{matomaki}
  K. Matomäki: \textit{The distribution of {$\alpha p$} modulo one},
  Math. Proc. Cambridge Philos. Soc. 147 (2009), 267--283.

\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}

\bibitem{polya} G. Pólya: \textit{\"{U}ber eine {A}ufgabe der
    {W}ahrscheinlichkeitsrechnung betreffend die {I}rrfahrt im {S}tra\ss
    ennetz}, Math. Ann. 84 (1921), 149--160.

\bibitem{vinogradov}
  I.M. Vinogradov: \textit{The method of trigonometrical sums in the
    theory of numbers}, Interscience Publishers, 1963.
\end{thebibliography}

\end{document}
