Riblet 2026 existence sidon set distinct distance constant

../

corollary_3_4: For every alpha <= 1/2 some infinite Sidon set has divergent sum of s^(-alpha), so the range alpha > 1/2 of Theorem 1.2 cannot be widened.

corollary_6_4: For g >= 2, h >= 2 and alpha > 1/h some B_h[g]-set attains the supremum, stated finite, of the sum of b^(-alpha) over B_h[g]-sets.

theorem_1_1: The generating functions of the Sidon sets form a compact subset of the analytic functions on the open unit disc, and suitably weighted integrals of them over [0, 1) attain their supremum.

theorem_1_2: For every alpha > 1/2 some Sidon set attains the supremum, over all Sidon sets, of the sum of s^(-alpha) over its elements.

theorem_1_4: Some Sidon set has reciprocal sum equal to the distinct distance constant, the supremum of the reciprocal sums of all Sidon sets.

theorem_5_1: The distinct distance constant lies between 2.16150003 (strict) and 2.247307; the introduction prints the lower bound as 2.1615001.

theorem_6_1: For every g >= 2 and alpha > 1/2 some B_2[g]-set attains the supremum, over all B_2[g]-sets, of the sum of b^(-alpha) over its elements.

theorem_6_2: For every real alpha some sum-free set of positive integers attains the supremum, over all such sets, of the sum of f^(-alpha) over its elements.

theorem_6_3: A continuous real function on a closed family of subsets of N, in the product topology, attains its supremum on that family.

---

Robin Riblet, Titien Schehr, Existence of a Sidon set for the distinct distance constant. arXiv preprint (2026). arXiv:2505.20851. The arXiv record names arXiv's non-exclusive distribution license (arXiv:2505.20851), every other right reserved. The copy read for this card is arXiv:2505.20851v2 (12 April 2026); version 1 is dated 27 May 2025.

The paper establishes a compactness property for Sidon sets and B_2[g]-sets: Theorem 1.1 (p. 3) shows that the generating functions of Sidon sets form a compact subset of the holomorphic functions on the unit disc, under uniform convergence on compact subsets, and the paper uses it to prove that suprema of reciprocal-power sums are attained. Theorem 1.2 (pp. 2, 5) gives, for every alpha > 1/2, a Sidon set S_alpha maximizing the sum of s^(-alpha) over Sidon sets, and Theorem 1.4 (pp. 2, 5) is the alpha = 1 case: there exists a Sidon set whose reciprocal sum equals the distinct distance constant DDC, a question the introduction cites from Salvia. Remark 1.3 and Corollary 3.4 (pp. 5, 10) show the range alpha > 1/2 is optimal: for alpha <= 1/2 some infinite Sidon set has divergent sum of s^(-alpha). Theorem 5.1, as stated on p. 11 and proved on pp. 11-12, gives 2.16150003 < DDC <= 2.247307; the lower bound comes from Kleinwaks's 1010-term Sidon set K extended by the elements 2^k max K, and the upper bound, improving the 2.24732646 the introduction credits to Taylor, from Taylor's approach combined with Lindstrom's cardinality bound (Lemma 5.2, p. 12). The introduction's statement of Theorem 5.1 (p. 2) prints the lower bound as 2.1615001, the value it reports from Kleinwaks's own remark that a greedy completion of K improves the bound; the proof concludes with 2.16150003. Section 6 transfers the method to B_2[g]-sets (Theorem 6.1, p. 13, for g >= 2 and alpha > 1/2) and to sum-free sets (Theorem 6.2, pp. 2, 14), and gives an elementary attainment theorem for continuous functions on closed subsets of P(N) (Theorem 6.3, pp. 3, 15), from which Corollary 6.4 (pp. 3, 15) gets maximizers over B_h[g] for g >= 2, h >= 2 and alpha > 1/h, the supremum stated finite; p. 15 asks whether the supremum is finite at alpha = 1/h. The introduction records Ruzsa's infinite Sidon set, of density of order n^(sqrt 2 - 2), as the best density currently known for Sidon sets. The paper does not mention problem 158. Its only lower bound on a counting function is Theorem 2.1 (p. 6): a Sidon set maximizing the reciprocal sum has |S cap [1, n]| > Cn^(1/4) for all n, for some C > 0.257, and positive upper limit of |S cap [1, n]|/n^(1/3).

Source: https://arxiv.org/abs/2505.20851.

Bears on. #158: Theorem 6.1 and Corollary 6.4 with g = 2 (and h = 2) concern the B_2[2] sets of the problem, but give maximizers of the sum of b^(-alpha) for alpha > 1/2, not a bound on the counting function; Corollary 3.4 shows that divergence of the sum of a^(-1/2), which a set with positive lower square-root density would have (the card's deduction, not the paper's), already occurs for some Sidon set. None of the results says anything about the lower limit of |A cap [1, N]|/N^(1/2), and the paper does not mention the problem.

Results. Labels and pages are those of v2.

• Theorem 1.1 (p. 3): the generating functions of Sidon sets form a compact set, and weighted integrals of them attain their supremum.
• Theorem 1.2 (pp. 2, 5): for every alpha > 1/2 a Sidon set attains the supremum of the sum of s^(-alpha) over Sidon sets.
• Theorem 1.4 (pp. 2, 5): some Sidon set has reciprocal sum equal to DDC.
• Corollary 3.4 (p. 10): for alpha <= 1/2 some infinite Sidon set has divergent sum of s^(-alpha).
• Theorem 5.1 (p. 11): 2.16150003 < DDC <= 2.247307; the introduction (p. 2) prints the lower bound as 2.1615001.
• Theorem 6.1 (p. 13): for g >= 2 and alpha > 1/2 a B_2[g]-set attains the supremum of the sum of b^(-alpha) over B_2[g]-sets.
• Theorem 6.2 (pp. 2, 14): for every real alpha a sum-free set attains the supremum of the sum of f^(-alpha) over sum-free sets.
• Theorem 6.3 (pp. 3, 15): a continuous real function on a closed subset of P(N), with the product topology, attains its supremum.
• Corollary 6.4 (pp. 3, 15): for g >= 2, h >= 2 and alpha > 1/h a B_h[g]-set attains the supremum, stated finite, of the sum of b^(-alpha) over B_h[g]-sets.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.