Obryant 2026 thickness infinite generalized sidon sets i

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Kevin O'Bryant, On the Thickness of Infinite Generalized Sidon Sets, I. arXiv preprint (2026). arXiv:2606.28651. The arXiv record (https://arxiv.org/abs/2606.28651, read 2026-10-02) names the Creative Commons Attribution 4.0 license. The copy read for this card is v3 (26 July 2026).

A set A of nonnegative integers is a g-Golomb ruler if every positive difference d has at most g representations as a - b with a, b in A; g = 1 gives Sidon sets. Theorem 1 proves that any g-Golomb ruler satisfies liminf A(n)/sqrt(n/log n) <= (2/sqrt(log 2))·sqrt(g), sharpening Cilleruelo's constant 8·sqrt(7) (about 21.2) down to about 2.4·sqrt(g) and extending from Sidon sets to g-Golomb rulers; Corollary 2 restates this as limsup a_n/(n^2 log n) >= (log 2)/(2g). Theorem 3 extends Krückeberg's counterpart: every g-Golomb ruler has limsup A(n)/sqrt(n) <= sqrt(g), and there exists one with limsup A(n)/sqrt(n) >= sqrt(g)/sqrt(2). The method is an energy argument over blocks of length N, averaged over shifts of the blocks, built on two finite counting lemmas (Lemmas 4 and 5) quoted from Caicedo, Martos and Trujillo. This is Part I of a three-part series; Part II treats B_h sets with h even and Part III odd h. For problem 158, the g-Golomb condition bounds repetitions of each difference, which for g > 1 neither implies nor follows from the B_2[g] sum-multiplicity condition ({0, 1, 2, 3} is a B_2[2] set but not a 2-Golomb ruler, and {0, 1, 3, 7, 9, 10} is a 2-Golomb ruler in which 10 has three representations), so the paper does not resolve 158 although its block-energy technique is directly adjacent.

Source: https://arxiv.org/abs/2606.28651.

Bears on. #158

Results to transcribe.

• Theorem 1: Every g-Golomb ruler A has liminf A(n)/sqrt(n/log n) <= (2/sqrt(log 2))·sqrt(g).
• Corollary 2: For an infinite g-Golomb ruler {a_1 < a_2 < ...}, limsup a_n/(n^2 log n) >= (log 2)/(2g).
• Theorem 3: Every g-Golomb ruler has limsup A(n)/sqrt(n) <= sqrt(g), and some g-Golomb ruler achieves limsup >= sqrt(g/2).
• Lemmas 4-5: Quoted upper and lower bounds of Caicedo, Martos and Trujillo on the size of g-Golomb rulers in [0, N).