Cost drag and the optimal holding period

(a) A strategy has gross per-trade Sharpe $e/\sigma_t = 0.2$ and cost ratio $c/\sigma_t = 0.139$. How many independent trades a year are needed for a net annual Sharpe above 1? (b) Suppose the gross edge grows like $e(h) = b\sqrt h$ with holding period $h$ (as for a signal that predicts a fraction of a random-walk move), $\sigma_t = \sigma_D\sqrt h$, $N = H/h$. Show that the annual Sharpe is maximised at $h^* = 4c^2/b^2$, and that at the optimum the gross edge is exactly twice the cost. Interpret.