What is the difference between "me asking" and "my asking"?
I think it's hard to see a semantic difference in your particular example, tufguy. But consider these:
(a) Would you mind me smoking
(b) Would you mind my smoking
The underlined phrase in (a) goes by the name of the ACC-ing construction in contemporary syntax. H. W. Fowler named it the "fused participle" construction about a hundred years ago, and he considered it illegitimate, at least in his dictionary of usage. (The Kings English
, published earlier, seems to allow for it in special cases -- e.g., a picture of him smoking
Whatever the true syntactic nature of the ACC-ing construction turns out to be -- the jury appears still to be out on exactly what it is (I analyze it as a nonfinite clause
) -- it differs from the syntax of the underlined phrase in (b), which goes by the name of the POSS-ing construction and is a noun phrase
or determiner phrase.
Although both (a) and (b) can arguably be substituted in a wide variety of cases, I find (a) ("Would you mind me smoking?") to be very natural as a substitute for "Would you mind it if I smoked?" (I'm not smoking now, but am considering it), or in a context where I am being contrasted with someone else, in which case "me" could even be emphasized: "Would you mind me
The POSS-ing construction ("Would you mind my smoking?") would, on the other hand, be the better choice, in my opinion, in a case where I already am smoking or am already in the habit of smoking. In the context I have in mind, the question is implicitly conditional and could be expanded as such: "Would you mind my smoking (if I went across the street)?"
P.S. This is just an example. I don't smoke (though I did for a long time) and would never recommend it to anyone.