Chris,

I believe you will find that this is a USA standard thread, a No3 x 48TPI UNC. As Engine Builder says, it would normally have a basic major dia of 0.0990, so 3/32 rod at 0.0938 is theoretically a little undersized, but you should get away with it. Tracey Tools ( no connection ) do a range of suitable UNC taps and split dies, with the carbon versions costing very little. For your information, die nuts are not meant for creating threads since they cannot be adjusted like a split die for fit, have less hook on the teeth as well as thicker legs. For these reasons, they are intended for cleaning worn or damaged threads only.

Regarding an M3 thread, this would normally have a basic major dia. of 3 mm ( 0.1181), thus making it unsuitable in this application.

Also, as stated, whilst there is no 3/32" ME thread, there is a 3/32" BSW x 48 TPI thread, which co-incidently has a basic major dia. of 0.0938, ie exactly the same as your rod, and again Tracey Tools have these available for relatively low cost.

I hope you will find the above info of interest.

Trevor.