Is this an idiom? When you say a study is "sparse on details" could you not just as well say "sparse with details" or "sparse for details"—they all make just as much sense to me. I'm trying to understand the role of the proposition on, if any. I understand that sometimes combination of words will become one, common grammatical unit and then be called idiomatic. This may be one of them.
Many constructions that end with the words "X on details"—where X indicates some type of deficiency—pair X with a corresponding positive word that characterizes a contrasting strength.
For example, a sentence might say this:
The speech was heavy on rhetoric [or hyperbole or enthusiasm] but light on details.
The political party was strong on generalities [or claims or atmosphere] but weak on details.
The program was long on indignation [or promises or vision] but short on details.
Or even this:
The coverage is rich in [or dense with or crowded with] general concepts but sparse on details.
I consider all of these wordings to be variations on a single rhetorical form in English—one that uses the trick of laying out a strength and then challenging its usefulness by noting a paired weakness. It may be that some specific wordings using the form rise to the level of idioms, due to being used so frequently. The most likely candidates in this regard, I think, are "heavy on Y but light on Z" and "strong on Y but weak on Z."
But the phrase "sparse on details" doesn't require the counterweight of (for example) "crowded with rhetorical flourishes" to function quite well in an English sentence; it just needs a setting where the thing being examined and described can reasonably be characterized as lacking in detail. In such a situation the phrase "sparse on details" doesn't seem to me to rise to the level of an idiom. It simply observes that details are few and far between, as one would expect from a literal reading of the word sparse.
On a more granular level, though, English does seem to strongly prefer the choice of on as the preposition in phrases of the form "X [preposition] details"—as my preceding examples have illustrated. And if patterns of preposition preference count as "idiomatic," then the preference for "X on details" qualifies.
As shown by Google Books, these two expressions are both idiomatic:
"sparse on details" About 774 results
"sparse with details" About 108 results
in that they are found in large enough numbers in published books to be accepted as regular expressions by the native speakers, but they are not idioms, IMO, for a search on:
fails to show in its results any idiom dictionary (my own test for an idiom :-))
Surely, I consider this definition of "idiom" as acceptable:
id·i·om n. 1. A speech form or an expression of a given language that is peculiar to itself grammatically or cannot be understood from the individual meanings of its elements, as in keep tabs on.
Of the two, as one can see, the first is 8x more frequently, a big reason to prefer it.
I can easily read:
sparse on details = sparse in terms of details
However I find
sparse with details
somehow ambiguous. Surely, it doesn't mean:
sparse, with details
because that might be even contradictory:-)