It seems like this could be uncharitably summarized as follows. "The usual definitions of fractal are rather narrow. The second author of the present paper has proposed a broader definition of fractal. In this paper we show that according to this definition curves such as the semicircle and the logarithmic spiral are 'fractals', but rather than concluding that the definition was a bad one we choose to conclude instead that those curves really are fractals."
Which is, of course, absurd. Am I missing something?
I agree with you - I can't pick any sense out of this paper. Note that they have not given a single example of a curve that is not a "fractal" by their definition. What would such a curve look like? A straight line is the obvious guess - but since all the "bends" are zero for a straight line it's not at all clear what their definition would even mean.
Intuitively, a fractal curve is something that has larger bends than expected yet their definition is something has a disproportionately many small bends. Why would we want to consider such a thing fractal? Frankly if I were to try to find a curve that was not fractal (under their definition), I would search amongst curves that are fractal (for the usual notion): how else would you get large bends even as you subdivide the problem?
An expansion of a definition is warranted if it is consistent with the original intent of the definition. I can see the log spiral as being construed as fractal although a semicircle is a bit off.
Which is, of course, absurd. Am I missing something?