Cilleruelo 2001 infinite b 2 g sequences

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conjecture_p1: The conjecture the introduction records, without attribution, that every infinite B_2[g] sequence has lower limit zero for A(x)/x^{1/2}, open even for g = 2, beside Erdős's theorem for Sidon sequences; the paper proves nothing on it.

theorem_1: Cilleruelo and Trujillo's construction, for each g >= 2, of an infinite B_2[g] sequence whose counting function has limsup A(x)/sqrt(x) equal to an explicit constant L_g, tabulated for g <= 8 and given by a formula for g >= 9; the proof as printed reaches the formula only for odd g >= 9.

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Javier Cilleruelo, Carlos Trujillo, Infinite B_2[g] sequences. Israel Journal of Mathematics 126 (2001), no. 1, 263-267. doi:10.1007/BF02784156.

Theorem 1 constructs, for every g >= 2, an infinite B_2[g] sequence A with limsup A(x)/x^(1/2) equal to an explicit constant L_g. The theorem (p. 2) tabulates L_g = (3/2)^(1/2), 3/2, (36/11)^(1/2), (9/2)^(1/2), (100/17)^(1/2), (27/4)^(1/2) and 8^(1/2) for g = 2, ..., 8 and sets L_g = (3/(2 sqrt 2)) (g-1)^(1/2) for g >= 9; for g = 2 this is an infinite B_2[2] sequence with limsup A(x)/x^(1/2) = (3/2)^(1/2). The construction extends any finite B_2[g] sequence A_0, with largest element x, by the translates B_p + cm + 2x for c in a finite set C_g in [0, u_g], where p is a prime with x^2 < p < 2x^2, m = p^2 - 1, and B_p has more than p - 4p^(1/2) elements and is a Sidon set modulo m (cut down from a modular Sidon set with p elements that the paper attributes to Chowla and Erdős, citing Halberstam and Roth); repeating the step gives the sequence, and L_g is the ratio |C_g|/(u_g + 1)^(1/2). This differs from Jia's approach, which only handled sequences with a bounded modular representation count. The abstract claims the formula for every g >= 2 with better values for small g, but the tabulated values equal the formula at g = 3, 5 and 7 and fall below it at g = 4. For g >= 9 the proof (p. 4) takes for C_g a set of Cilleruelo, Ruzsa and Trujillo that for even g gives only (3g-4)/(2(2g-3)^(1/2)), slightly below the formula (13/17^(1/2), about 3.153 against 3.182, at g = 10), so in the version read the proof reaches the stated L_g only for odd g >= 9 (a check made for this card). The values improve on Kolountzakis's infinite B_2[2] sequence with limsup A(x)/x^(1/2) = 1. The introduction records the state of the art for problem 158: Erdős proved liminf A(x)/x^(1/2) = 0 for every infinite Sidon sequence, while for g > 1 the same vanishing for infinite B_2[g] sequences is only conjectured and is open already at g = 2. The paper therefore only bears on the limit superior; the limsup construction supplies no positive liminf and so does not refute #158.

Source: https://doi.org/10.1007/BF02784156. The copy read for this card is an author-typeset version, which prints no notice; the version of record's Springer article page shows only the site footer "© 2026 Springer Nature" and names no license (https://link.springer.com/article/10.1007/BF02784156, read 2026-10-02), and does not govern that version; the term is unstated.

Bears on. #158: Theorem 1 at g = 2 gives an infinite set with at most two solutions of a + a' = n, a <= a', and limsup A(x)/x^(1/2) = (3/2)^(1/2); the problem asks about the liminf, on which the paper proves nothing, and the conjecture on p. 1 that every infinite B_2[g] sequence has liminf A(x)/x^(1/2) = 0, recorded as open even for g = 2, asserts at g = 2 the affirmative answer to the problem's question. #329: the problem asks for the largest limsup A(x)/x^(1/2) of a Sidon set (g = 1); Theorem 1 concerns g >= 2, a wider class of sets, and gives no bound for Sidon sets. The problem's site lists the paper among its references.

Results.

• Theorem 1 (p. 2): For every g >= 2 there is an infinite B_2[g] sequence with limsup A(x)/x^(1/2) = L_g, where L_g is tabulated for 2 <= g <= 8 and equals (3/(2 sqrt 2))(g-1)^(1/2) for g >= 9; in the version read, the proof reaches this formula only for odd g >= 9 (for even g its set C_g gives (3g-4)/(2(2g-3)^(1/2))). Its case g = 2 is an infinite B_2[2] sequence with limsup A(x)/x^(1/2) = (3/2)^(1/2), improving Kolountzakis's value 1.
• Conjecture, p. 1: the conjecture, recorded without attribution, that every infinite B_2[g] sequence has liminf A(x)/x^(1/2) = 0, open already for g = 2; Erdős proved it for Sidon sequences (g = 1).

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