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 definiti

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?