From a name point of view I would say you're fine. If your actual logo, branding etc is similar too, then you might be in trouble, for example using the same colours and fonts etc. iAAA is different from I AAA THAT. What are the companies services? Again if they're different it's easier to differentiate. I guess the real point in copy write is preventing people knocking off other's ideas. It's no secret Apple was taken from the Beatles Apple Music, but was given the green light to sell computers and not music, however, with the iPod I don't know if that changed or anything.
It's like saying Apple Computers infringes on Apple Music. It has part of the same name, but they're two separate companies selling two different products.
Best bet above is speak to a lawyer, or better yet, do what Steve did and contact IAAA and ask if they're comfortable with the new name. Because at the end of the day it will be them suing you, and you paying to defend yourself.