Cilleruelo 2015 sidon sets asymptotic bases

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theorem_1_1: For every sufficiently large N the cyclic group Z_N contains a Sidon set that is a basis of order 3 in Z_N, the modular version of Erdős's conjecture on Sidon bases of order 3.

theorem_1_2: There is a sequence of positive integers in which every integer has at most two representations as a sum of two terms and every large integer is a sum of three terms.

theorem_1_3: For every positive epsilon there is a Sidon sequence of positive integers in which every large n is a sum of four terms, one of them at most n to the power epsilon.

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Javier Cilleruelo, On Sidon sets and asymptotic bases. Proceedings of the London Mathematical Society 111, no. 5 (2015), 1206-1230. arXiv:1304.5351, doi:10.1112/plms/pdv050.

The paper attacks Erdos's Conjecture 1.1, that some infinite Sidon sequence is an asymptotic basis of order 3, from three directions. Theorem 1.1 proves the modular version: for all large N the cyclic group Z_N contains a Sidon set that is a basis of order 3, using a result of Granville, Shparlinski and Zaharescu on the distribution of points from curves over F_p in the s-dimensional torus. Theorem 1.2 proves by the Erdos-Renyi probabilistic method that some B_2[2] sequence of positive integers is an asymptotic basis of order 3, so the minimal g Erdos asked about is at most 2. Its proof does not use the space S(gamma) itself, where the events x in A are independent with P(x in A) = x^{-gamma}, but the variant of Definition 3, which admits only integers x > m lying in the residue classes of a modular Sidon basis of Z_N from Theorem 2.1; it takes gamma = 7/11 (any gamma in (5/8, 2/3) would do) and then deletes every element that is a summand in some sum with three distinct representations. Definition 2 introduces asymptotic bases of order h + epsilon, and Theorem 1.3 shows that for every epsilon > 0 there is a Sidon sequence in which every large n is a sum of four elements one of which is at most n^epsilon. Conjecture 1.1 is problem 157, which the paper does not settle; the B_2[2] sequence of Theorem 1.2 lies in the class of problem 158, but the paper gives no bound on its counting function at the scale N^(1/2).

Source: https://arxiv.org/abs/1304.5351. The copy read for this card is arXiv:1304.5351v2 (24 April 2013), titled "Sidon basis", not the journal version; the labels and pages cited here are that version's. The arXiv record names arXiv's non-exclusive distribution license (arXiv:1304.5351), every other right reserved.

Bears on. #157: the problem is the paper's Conjecture 1.1, an infinite Sidon set that is an asymptotic basis of order 3. The paper proves three approximations and does not settle it: Theorem 1.1 (p. 1) is the analogue in Z_N for all large N, Theorem 1.2 (p. 2) relaxes the Sidon condition to B_2[2], and Theorem 1.3 (p. 2) keeps the Sidon condition and uses four summands, one at most n^epsilon. #158: the problem's sets, infinite with at most two solutions of a + b = n with a <= b, are the infinite B_2[2] sequences of the paper's Definition 1. Theorem 1.2 (p. 2) constructs one that is an asymptotic basis of order 3; the paper states no bound on its counting function at the scale N^(1/2), and it does not bear on the liminf the problem asks about.

Results. Labels and pages are those of arXiv:1304.5351v2 (pp. 1--32).

• Theorem 1.1 (p. 1; proof Sections 2.1--2.2, pp. 6--10): for all sufficiently large N, Z_N contains a Sidon set that is a basis of order 3 in Z_N.
• Theorem 1.2 (p. 2; proof Section 4, pp. 14--17): there exists a B_2[2] sequence of positive integers that is an asymptotic basis of order 3, so the least g in Erdős's question is at most 2.
• Theorem 1.3 (p. 2; proof Section 5, pp. 18--21): for every epsilon > 0 there is a Sidon basis of order 3 + epsilon in the sense of Definition 2 (p. 2): every large n is a sum of four elements of the sequence, one of them at most n^epsilon.

Theorem 2.1 (p. 4) and Corollary 2.1 (p. 5), which give, in infinitely many cyclic groups Z_N, Sidon sets over which every element is a sum of three, respectively four, pairwise distinct elements, are recorded within the proof pointers of Theorems 1.1 to 1.3 and have no pages of their own. The probabilistic background on p. 2 is likewise recorded here only: in S(gamma), for gamma > 3/4 almost all sequences become Sidon after removing finitely many elements, and by Erdős and Tetali, for gamma < 1 - 1/h almost all are asymptotic bases of order h, which for gamma in (3/4, 4/5) gives Sidon bases of order 5, the argument of Kiss.

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