Do you mind if I open the window? (OALD)
If I use do you mind my opening the window or do you mind me opening the window, are both plausible or not?
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All three are fine in everyday use. "Do you mind me opening the window" is technically questionable (because pedants will say that it's really asking if they would prefer somebody else to do it), but it is the sort of "mistake" that will make you sound even more like a native speaker, because it's so common.