Is it r-g or f r-g?

One measure of inequality for Piketty is the capital to income ratio β =C/I. If the rate of return on capital is r, not all the return rC is invested next year but only a fraction of it frC. which is the increase in capital that is used to calculate the  β for next year. This leads to the formula β_(t+1)/β_t=(1+f_t r_t)/(1+g_t) which helps to determine the growth of β in the long run. So it seems that what matters is fr-g rather than r-g. To see what f is like, if s is the national savings rate frC =sI or frβ=s or β=s/fr. So if β does not explode, we should have fr and g nearly equal and this leads the formula β=s/g in the long run, the so called Harrod-Domar-Solow formula. In any case, the emphasis on fr-g instead of that on r-g would have made Piketty's arguments clearer.
P.S. Calculations of this type are in a paper of Thomas Piketty and Gabriel Zucman  http://behl.berkeley.edu/files/2013/02/Piketty-Zucman_WP2013-10.pdf