I find that often researchers in math do not define every piece of notation that they use, however, I am not sure what criteria they use to deem that a definition should be left out, or should be included.
For example, in Terence Tao's paper, he did not provide a definition of the gamma function. But the gamma function is standard, so there shouldn't be any problems.
In these other papers on optimization, for example, the researchers uniformly do not define what they mean by $\Omega, \omega, O$, or $o$ and assume that they are well known (even though I can think of conflicting definitions in CS versus math, see chapter 1 of CLRS versus Luenberger and Ye's optimization text)
I am sure there are many other examples. Now I am beginning to learn how to communicate in math, but I am not sure what I should define, what I should leave out. And if I define everything, it soon becomes a rabbit hole, but if I don't define, I get yelled at for being fast and loose and imprecise.
How do you determine what to define?