Why do you assume there is an AE/BE difference?
This is a common misapprehension when different usages are noticed.
True, there are many AE/BE differences, but there are also many other differences that appear in both AE and BE.
Crude Google measures show that fixation with and fixation on are approximately equally represented across the web and within UK sites. World
about 1,100,000 for "fixation with about 1,080,000 for "fixation on UK
about 69,000 for "fixation with about 49,400 for "fixation on
That suggests to me that we use with/on equally and there is no difference between AE and BE.
I assume there is a an AE/BE difference because I came across the two versions on the Internet some time ago. Then, while browsing through an English grammar book (the author of which is British) I found the answer with.