“Thus he argues that video surveillance by or with the consent of a government informant constitutes a search within the gambit of the Fourth Amendment’s protections, and a warrant is therefore necessary to legally conduct such video surveillance.” From United States v. Brathwaite (5th Cir. 2006).
A Westlaw ALLCASES search for “within the gambit of” uncovers 108 cases, and a quick eyeballing suggests that all or nearly all really do mean “within the ambit of.” I’ve checked the original of Brathwaite to confirm that “gambit” is in the printed reporter and not just in the Westlaw version, but I haven’t checked the other cases.
I’m pleased to report, though, that “within the gambit” hasn’t yet become standard. “Within the gambit of” & date(> 1/1/2009) yields 7 results; “within the ambit of” & date(> 1/1/2009) yields 1810.