There is a bit of confusion in my mind about this style of AS. I have seen a few that I have been concerned are Chinese, they are irregular in the spirals and not finished well at the top. Not sure if this style has been copied but there are a fair few of them about (if you are looking). With the age and old label on the bottom of yours I am sure yours is genuine.
Regarding the signature on the bottom. This was a tourist piece for AS and used in various other designs, lamps, bottles etc., he would not have signed them as he signed only significant pieces.
This is my perfume bottle.
Thanks for the $27 note - I just hated you before, now you have become the Arch Enemy (lucky I don't hold a grudge).