You're right that the g1 and g2 in section 2 are also scaled in addition to being shifts of f. I plan to clarify this in the text.

This work is inspired by the self-similarity of fractals, which is in some cases primarily an intuitive or informal idea: for example, the Mandelbrot set is considered self-similar, but I'm not aware (despite searching) of any simple automorphisms of the Mandelbrot.

From the first sentence of the paper: “I explore functions f that can be written as a sum f=g1+g2 where g1 and g2 are shifted and possibly reflected versions of each other, both strongly resembling the original function f.”

Note, not f but “both strongly resembling the original function f.”

The limit of `g_i / f` is also not 1. I think he's just looking for convergent functions.