Obryant 2026 thickness infinite generalized sidon sets ii

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Kevin O'Bryant, On the Thickness of Infinite Generalized Sidon Sets, II. arXiv preprint (2026). arXiv:2607.23795. The arXiv record (https://arxiv.org/abs/2607.23795, read 2026-10-02) names the Creative Commons Attribution 4.0 license. The copy read for this card is v1 (26 July 2026).

For A a B_h set (all h-fold sums with nondecreasing summands distinct) and A(n) = |A intersect [0, n)|, Theorem 1 proves that for every even h, liminf A(n)/(n/log n)^{1/h} <= ((pi/log 2)·Gamma(1 + h/2)^2/Gamma(1 + 1/h)^h)^{1/h}. Corollary 2 restates this as limsup a_n/(n^h log n) >= ((log 2)/pi)·h·Gamma(1 + 1/h)^h/Gamma(1 + h/2)^2. Chen (Acta Arith. 1993) proved finiteness of this liminf without a constant; the contribution here is the explicit constant, obtained by an energy argument over blocks of length N averaged over shifts, together with a new lower bound (Lemma 8) on the size of the (h/2)-fold sumset, whose integral over a simplex produces the Gamma factors. At h = 2 the theorem reduces to the ordinary Sidon case and recovers the corresponding result of Part I. The best complementary construction cited is Cilleruelo's B_h set with G(n) = n^{sqrt((h-1)^2+1)-(h-1)+o(1)}. For odd h the paper gets only the bound for h - 1, since every B_h set is a B_{h-1} set. For problem 158 the review note records that the paper treats unique B_h sets, not bounded-multiplicity B_2[2] sets, and constructs no dense one.

Source: https://arxiv.org/abs/2607.23795.

Bears on. #158

Results to transcribe.

• Theorem 1: For even h and any B_h set A, liminf A(n)/(n/log n)^{1/h} <= ((pi/log 2)·Gamma(1+h/2)^2/Gamma(1+1/h)^h)^{1/h}.
• Corollary 2: For an infinite B_h set with h even, limsup a_n/(n^h log n) >= ((log 2)/pi)·h·Gamma(1+1/h)^h/Gamma(1+h/2)^2.
• Remark (h = 2): At h = 2 Theorem 1 reduces to a special case of the g-Golomb ruler result of Part I.