Carter 2025 diameter finite sidon sets

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Carter, D. and Hunter, Z. and O'Bryant, K., On the diameter of finite Sidon sets. Acta Math. Hungar. 175 (2025), no. 1, 108--126. DOI: 10.1007/s10474-024-01499-8. The copy read for this card is the arXiv version; page numbers below are that version's.

Writing b_infinity for the lim sup of (k^2 - diam(A_k))/k^{3/2} over minimum-diameter k-element Sidon sets, Singer's construction and Erdos-Turan give 0 <= b_infinity <= 2; Balogh-Furedi-Roy improved this to 1.996 and O'Bryant to 1.99405. This paper proves b_infinity <= 1.96365 (Section 3), a comparatively large improvement, equivalently that a Sidon set of diameter n has at most n^{1/2} + 0.98183 n^{1/4} + O(1) elements; Section 2 gives a hand-verifiable proof of the weaker b_infinity <= 1.99058. The method is a simplified exploitation of the Erdos-Turan argument through the Erdos-Turan Sidon Set Equality (Theorem 1.1), which expresses diam(A) exactly in terms of the slack quantities S(A,T) and V(A,T); the strongest bound is conceptually simple but computationally heavy and relies on substantial computer assistance. Section 4 extends the analysis to g-thin Sidon sets (g-Golomb rulers), proving diam(A) >= g^{-1}k^2 - (2-eps)g^{-1}k^{3/2} - O(k) with eps >= 0.02 g^{-2} (stated as eps

= 1/(50g^2), constant not optimized). The results bear directly on problem 30, the sharp size of a Sidon set in {1,...,N} and the second-order term in the Erdos-Turan upper bound.

Source: https://arxiv.org/abs/2310.20032. The arXiv record (https://arxiv.org/abs/2310.20032, read 2026-10-02) names the Creative Commons Attribution 4.0 license.

Bears on. #30

Results to transcribe.

• Theorem 3.3 (p. 12), the main theorem: b_infinity <= 1.96365: a k-element Sidon set has diameter at least k^2 - 1.96365 k^{3/2} - O(k), equivalently R(n) <= n^{1/2} + 0.98183 n^{1/4} + O(1), where R(n) is the largest size of a Sidon set in [0,n); proof uses substantial computer assistance. The theorem as printed reads diam(A) <= k^2 - 1.96365 k^{3/2} - O(k); the abstract and Section 1 state the lower bound, so the printed inequality sign is a misprint.
• Theorem 2.1 (p. 4), the hand-verifiable bound: a k-element Sidon set has diameter at least k^2 - 1.99058 k^{3/2} - O(k), so b_infinity <= 1.99058, proved without computer assistance, already improving on Balogh-Furedi-Roy (1.996) and O'Bryant (1.99405).
• Theorem 1.1 (p. 2, ETSSE, from O'Bryant [OBr22]): For a finite Sidon set A and positive integer T, diam(A) = |A|^2 T^2 / (T(T+|A|-1) - (2S(A,T)+V(A,T))) - T, the exact identity driving the argument.
• g-thin Sidon sets (Section 4; stated on p. 2, Theorem 4.4 on p. 15): A g-thin Sidon set with k elements has diam(A) >= g^{-1}k^2 - (2-eps)g^{-1}k^{3/2} - O(k) with eps >= 1/(50 g^2).