In the introduction of a paper submission I had the sentence
We show that, for a rather general set of probability distributions, the thickness of a random graph is asymptotically almost surely at most polylogarithmic.
(Sentence 1).
As usual, an event depending on a parameter is said to hold asymptotically almost surely if the probability of success goes to 1 in the limit →∞ ("Topics in Random Matrix Theory", Terence Tao, 2012). The mathematical meaning of Sentence 1 is this:
We show that, for a rather general set of probability distributions, there is some positive integer constant $c$ such that we have $\lim_{n→∞} \mathrm{Prob}[\mathrm{thickness}(\mathrm{random}\ \mathrm{graph}\ \mathrm{of}\ \mathrm{size}\ n) ⩽ \ln^c n] = 1$.
Let us leave aside the mathematical questions such as what thickness means (simply imagine some function from graphs to natural numbers), how to measure the size of a graph (imagine the number of nodes), what "a rather general set of probability distributions" means, and how to encode it into $\mathrm{Prob}$.
The formula is ok, but it has one catch: we don't wish to use any formulas in the introduction, so, a plain English version is needed.
An English teacher (who is not a mathematician) disliked "at most" in the plain English version (similarly to http://english.stackexchange.com/questions/198236/should-at-most-four-have-commas) and wanted to have commas around it or rewrite it. The two options at my disposition now are these:
We show that, for a rather general set of probability distributions, the thickness of a random graph is asymptotically almost surely, at most, polylogarithmic.
(Sentence 2)
We show that, for a rather general set of probability distributions, the thickness of a random graph is asymptotically almost surely bounded from above by a polylogarithmic function.
(Sentence 3)
Sentence 2 looks unnatural to me, while Sentence 3 looks simply way too long to me. Has the teacher overcorrected? Is there any better way to express the above formula in plain English?
PS. I've asked the question at English.SE, but the folks there seem not to know better so far.
Mathematical English is not quite the same as non-mathematical English. Your first formulation seems perfectly fine to me. Putting in the commas would only confuse mathematicians.
Actually, if you use the definition of polylogarithmic that is common in complexity theory (see definition (2) here), you could just omit the "at most" and say it is asymptotically almost surely polylogarithmic.