Nonstandard analysis: Typesetting `^*` depending on context

Question

In nonstandard analysis, the nonstandard extension operator * is used frequently. Typesetting it is not straightforward, though, because the placement of * depends on the symbol to follow. E.g., the nonstandard extension of the reals is written

^*{\mathbb{R}}

whereas the nonstandard extension of a real function X is typically typeset

^*\!{X}

If the negative space \! is not included, the * and the X are spaced too far apart.

I'd like to separate style from content in my LaTeX, but it's not clear how I would define a macro that expanded appropriately given the context. It would seem that I would need two macros—one where I needed a negative space, and another when I didn't. But this seems not much better than adding \! to the code. I've seen posts suggesting the tensor package, but the space is not correct.

EDIT: I've accepted an answer that allows one to specify the appropriate spacing for particular letters. A LuaLateX version is a more flexible version of this idea. The automatic approach is impressive and creative, but doesn't deliver the quality necessary for a professionally typeset document. I'm now inclined to think that an automatic approach is unlikely to suffice without detailed knowledge of the underlying font.

Answer 1

Until someone comes up with a nice solution, a brute-force one could be:

\documentclass{scrartcl}
\usepackage{xparse,dsfont}

\ExplSyntaxOn
\NewDocumentCommand \nsext { m }
 {
  {\vphantom{#1}}
  \sp
   {
    *
    \str_case:nn {#1}
     {
      { X } { \mskip-3mu }
      { A } { \mskip-6mu }
     }
   }
  #1
 }
\ExplSyntaxOff

\newcommand*{\R}{\mathds{R}}

\begin{document}
$\nsext\R \quad \nsext X \quad \nsext V \quad \nsext A$
\end{document}

You just need to add a pair inside the \str_case:nn second argument if you want to add a new letter and the according space to remove.

Answer 2

REVISED APPROACH

Comments from the OP indicated that my original solution, while perhaps nice to look at, relied on changing the math font to ptmx, which was not acceptable. So the issue seemed to be that the math kerning of the ptmx font was OK, but that of ComputerModern (CM) was inadequate for the current task.

With that in mind, I decided to declare the ptmx math alphabet separately, and use it only for positioning of the CM glyphs. EDITED to declare a new math alphabet. Then, when I am stacking the * over/before the given argument, I use the \mathptmx version of the argument (that I just declared) to govern the offset from the right hand.

To account for arguments that are not pure alphabetic glyphs, I start out with a catcode test. In this MWE below, you see my approach on the top line, compared with the raw ComputerModern construction of $^*<letter>$ on the second line.

EDITED (8/2016) to work in subscript math styles, per an e-mail request of a reader. For this, I use the \ThisStyle{...\SavedStyle...} feature of the scalerel package to import the math style to places where it would otherwise be lost. RE-EDITED to \leavevmode to handle use-case in \substack.

\documentclass{article}
\usepackage{amssymb,stackengine,xcolor,scalerel,mathtools}
\stackMath
\def\nsa#1{\leavevmode\ThisStyle{%
\def\stackalignment{r}\def\stacktype{L}%
\ifcat A#1
  \mkern-6.5mu\stackon[0pt]{\SavedStyle\phantom{f}#1}  
    {\SavedStyle^*\mkern-1.1mu\phantom{\mathptmx{#1}}}%
\else
  \mkern-4mu\stackon[0pt]{\SavedStyle\phantom{f}#1}  
    {\SavedStyle^*\mkern-1.7mu\phantom{#1}}%
\fi
}}
\def\R{\mathbb{R}}
\DeclareMathAlphabet{\mathptmx}{OML}{ztmcm}{m}{it}
\parskip 1ex
\begin{document}
\centering
$(\nsa\R) ~ (\nsa V) ~ (\nsa X) ~ (\nsa A) ~ (\nsa M)$

vs.

$(^*\R) ~ (^*V) ~ (^*X) ~ (^*A) ~ (^*M)$

\hrulefill

Other cases requiring EDIT to \textbackslash nsa:

$(x_n)_{n\in\nsa{\mathbb N}}$. 

$\bigcup_{\substack{U\subseteq X\\ \nsa U\subseteq \mathrm{Fin}(\nsa X)}}$
\end{document}

ORIGINAL APPROACH (ptmx math)

This tries to align the * approximately where the right end of an f might be. The first row shows the kerning I was trying to emulate (the model); the second row shows the implemented macro; while the third row shows how the macro succeeds at its goal (the method, with * overlaying right end of f)

\documentclass{article}
\usepackage{amssymb,mathptmx,stackengine,xcolor}
\stackMath
\def\nsa#1{\def\stackalignment{r}\def\stacktype{L}%
  \mkern-1mu\stackon[0pt]{\mkern-2mu\phantom{f}#1}{^*\mkern-1.7mu\phantom{#1}}}
\def\R{\mathbb{R}}
\begin{document}
$ f\R ~fV ~fX ~fA$ The model

$\nsa\R ~ \nsa V ~ \nsa X ~ \nsa A$ The macro

\def\nsa#1{\def\stackalignment{r}\def\stacktype{L}%
  \mkern-1mu\stackon[0pt]{\color{cyan}\mkern-2mu f#1}{^*\mkern-1.7mu #1}}
$\nsa\R ~ \nsa V ~ \nsa X ~ \nsa A$ The method
\end{document}

Answer 3

Here's a LuaLaTeX-based solution, which sets up a Lua function that adjusts the space between the asterisk and the subsequent letter, where the adjustment amount depends on the shape of the letter.

The code defines LaTeX macro named \nsx (short for "nonstandard extension") that prefixes an asterisk to the macro's argument -- typically an uppercase letter; the default spacing adjustment between the asterisk and the letter is -4mu. (A negative thinspace, \!, equals -3mu.) The code next sets up a Lua function that overrides the default adjustment amount for selected letters.

See the table below for the adjustment amounts I've been able to come up with for the 26 uppercase letters of the Latin alphabet as well as for \mathbb{R} and \Gamma. Note that these adjustment amounts are optimized for the "Computer/Latin Modern" math fonts. Other font families will probably require different adjustment amounts.

% !TEX TS-program = lualatex
\documentclass{article}

\newcommand\nsx[2][4]{{}^{*}\mkern-#1mu#2}  % default neg. space: 4mu

\usepackage{amsfonts,array,booktabs} % just for this example

\usepackage{luacode,luatexbase}
\begin{luacode}
function adjust_ns ( line )
    if string.find ( line, "\\nsx" ) then
       line = string.gsub ( line, "\\nsx{([AJ])}", "\\nsx[6.5]{%1}" )
       line = string.gsub ( line, "\\nsx{([X])}", "\\nsx[4.5]{%1}" )
       line = string.gsub ( line, "\\nsx{([SZ])}", "\\nsx[3.5]{%1}" )
       line = string.gsub ( line, "\\nsx{([CGOQUVW])}", "\\nsx[2.5]{%1}" )
       line = string.gsub ( line, "\\nsx{([Y])}", "\\nsx[1.5]{%1}" )
       line = string.gsub ( line, "\\nsx{([T])}", "\\nsx[1]{%1}" )
       line = string.gsub ( line, "\\nsx{\\mathbb{R}}", "\\nsx[1.5]{\\mathbb{R}}" )
       line = string.gsub ( line, "\\nsx{\\Gamma}", "\\nsx[2]{\\Gamma}" )
    end
    return  line
end
luatexbase.add_to_callback ( "process_input_buffer", adjust_ns, "adjust_ns" )
\end{luacode}

\begin{document}

\noindent
\begin{tabular}{@{} >{$}l<{$} l @{}}
$Letter$ & Adjustment (in ``mu'')\\
\midrule
\nsx{B}\nsx{D}\nsx{E}\nsx{F}\nsx{H}\nsx{I}\nsx{K}\nsx{L}\nsx{M}\nsx{N}\nsx{P}
\nsx{R} & 4 (default)\\
\nsx{A}\nsx{J} & 6.5\\
\nsx{X} & 4.5\\
\nsx{S}\nsx{Z} & 3.5\\
\nsx{C}\nsx{G}\nsx{O}\nsx{Q}\nsx{U}\nsx{V}\nsx{W} & 2.5 \\
\nsx{Y} & 1.5 \\
\nsx{T} & 1 \\
\nsx{\mathbb{R}} & 1.5 \\
\nsx{\Gamma} & 2 \\
\end{tabular}
\end{document} 

Answer 4

This code also recognizes some types, based on the macro \binrel@: binary operation and relations (no operators, though).

\documentclass{article}
\usepackage{amsmath}
\usepackage{amssymb}

\makeatletter
\DeclareRobustCommand{\nsext}[1]{%
  \binrel@{#1}% compute the type
  \binrel@@{%
    {\vphantom{#1}}^*% the asterisk at the proper height
    \kern-\scriptspace % remove the script space
    \csname mkern@\detokenize{#1}\endcsname % additional kerning
    {#1}% the symbol
  }%
}
\newcommand{\defineextkern}[2]{%
  \@namedef{mkern@\detokenize{#1}}{\mkern#2}%
}
\makeatother

% define some additional kerning
\defineextkern{X}{-3mu}
\defineextkern{\in}{-2mu}

\begin{document}

$x\nsext{\in}\nsext{\mathbb{R}}$

$\nsext{X}_{x\nsext{\in}\nsext{X}}$

\end{document}