Cilleruelo 2010 probabilistic constructions b 2 g sequences

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theorem_1: Cilleruelo's criterion that probabilities p_n in [0,1] with sum up to t growing faster than log t and satisfying the dyadic condition (2) for a given g yield a B_2[g] sequence inside the support of (p_n) whose counting function is asymptotic to the sum of the p_n up to x.

theorem_2: Cilleruelo's theorem that for every positive integer g there is a B_2[g] sequence with a_k <= k^(2+1/g) (log k)^(1/g+o(1)) as k tends to infinity, improving the exponent 2 + 2/g of Erdős and Rényi.

theorem_3: Cilleruelo's theorem that for every positive integer g there is a B_2[g] sequence of squares with a_k < k^(2+1/g) (log k)^(kappa_g) for all k >= 2, where kappa_g is a positive constant depending on g.

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Javier Cilleruelo, Probabilistic constructions of B_2[g] sequences. Acta Mathematica Sinica, English Series 26 (2010), 1309–1314. doi:10.1007/s10114-010-8272-7.

Cilleruelo sharpens the Erdos-Renyi probabilistic construction of infinite B_2[g] sequences, sequences in which every integer has at most g representations as a sum x + y of two members with y <= x. Theorem 1 is a general criterion: for probabilities (p_n) satisfying a divergence condition and a dyadic-block convergence condition (2), there is a B_2[g] sequence A inside {n : p_n > 0} with A(x) ~ sum of p_n over n <= x. Theorem 2 deduces that for every g there is a B_2[g] sequence with a_k <= k^{2+1/g} (log k)^{1/g+o(1)}, improving the exponent 2 + 2/g of Erdos and Renyi (1960), and Theorem 3 gives the same exponent inside the perfect squares, a_k < k^{2+1/g}(log k)^{kappa_g}. The technical novelty is that the exceptional set, the members of the random set that take part in a sum with more than g representations, is not finite but has only a few elements in each dyadic interval, so an alteration step removes a few bad elements per block instead of finitely many overall. For problem 158 this is the direct fixed-g construction: exponent 2 + 1/g corresponds to counting exponent g/(2g+1) up to logarithms, so g = 2 gives 2/5, weaker than the Sidon exponent sqrt(2) - 1 and therefore not the unrestricted baseline for the problem.

Source: https://doi.org/10.1007/s10114-010-8272-7. The copy read for this card is the author's preprint dated June 3, 2008, which prints no notice; the version of record's Springer article page shows "© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg" and names no license (https://link.springer.com/article/10.1007/s10114-010-8272-7, read 2026-10-02), and does not govern that manuscript; the term is unstated.

Bears on. #158: Theorem 2 with g=2 gives an infinite B_2[2] set, representations counted with y <= x as the problem counts them, whose counting function is at least x^(2/5) up to logarithmic factors; that is far below N^(1/2), so the theorem gives no counterexample, and its exponent is weaker than the exponent sqrt(2) - 1 of infinite Sidon sets. Theorem 3 gives the same exponent inside the squares. The paper does not mention the problem.

Results. Labels and pages are those of the author's preprint dated June 3, 2008 (pp. 1-7); the journal's pagination differs.

• Theorem 1 (p. 2): if (p_n) in [0,1] satisfies (1/log t) sum_{n <= t} p_n -> infinity and the dyadic convergence condition (2) for a given g, then there is a B_2[g] sequence A inside {n : p_n > 0} with A(x) ~ sum_{n <= x} p_n.
• Theorem 2 (p. 2): for every positive integer g there is a B_2[g] sequence with a_k <= k^{2+1/g}(log k)^{1/g+o(1)} as k -> infinity, improving the Erdos-Renyi exponent 2 + 2/g.
• Theorem 3 (p. 2): for every positive integer g there is a B_2[g] sequence of squares with a_k < k^{2+1/g}(log k)^{kappa_g} for all k >= 2, where kappa_g is some positive constant depending on g.

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