Pliego 2024 erdos turan conjecture growth b 2

../

conjecture_1_1: The Erdős-Turán conjecture in the form Pliego states it: no set of natural numbers has between 1 and g unordered representations of every large integer as a sum of two elements, for any fixed g at least 2; the paper proves nothing on it.

conjecture_1_2: Pliego's prediction that for every eps > 0 some Sidon sequence represents every large integer as a sum of three of its elements with one summand at most n^eps, the Sidon analogue of his Corollary 1.1; the paper proves nothing on it.

conjecture_p2: The stronger conjectural statement Pliego attributes to Erdős and Fuchs, that for every g at least 2 each B_2[g] sequence has lower limit zero for its counting function over the square root; it would imply the Erdős-Turán conjecture, and the paper proves nothing on it.

corollary_1_1: The form of Pliego's Theorem 1.1 stated with a power: for 0 < eps < 1 and every integer g > 1/eps there is a B_2[g] sequence in which every large integer is a sum of three elements, one of them at most n^eps.

corollary_1_2: The counting part of Pliego's Theorem 1.1 on its own: for each g at least 2 some B_2[g] sequence of positive integers has at least a constant times x^{g/(2g+1)} elements up to x for large x, without a logarithmic or x^{o(1)} loss.

corollary_1_3: The case g = 2 of Pliego's Theorem 1.1, stated for the Erdős-Nathanson question on B_2[g] asymptotic bases of order 3: some B_2[2] sequence represents every large n as a sum of three elements, one of them at most a constant times n^{1/2}(log n)^{5/2}.

theorem_1_1: Pliego's main theorem: for every integer g at least 2 there is a B_2[g] sequence whose counting function is at least a constant times x^{g/(2g+1)} and in which every large integer is a sum of three elements, one of them at most a constant times n^{1/g}(log n)^{2+1/g}.

---

Javier Pliego, On the Erdős-Turán conjecture and the growth of B_2[g] sequences. arXiv preprint (2024), read in the version arXiv:2405.04154v1 (7 May 2024). The arXiv record names arXiv's non-exclusive distribution license (arXiv:2405.04154), every other right reserved.

Here r_A(x) counts the unordered pairs {a_1, a_2} of elements of A with a_1 + a_2 = x, so a sum a + a counts once (p. 1; the abstract says b_1 <= b_2), and A is B_2[g] when r_A(m) <= g for every m. Theorem 1.1 builds, for every integer g >= 2, a B_2[g] sequence A such that every large n can be written as a_1 + a_2 + a_3 with a_i in A and a_3 << n^{1/g}(log n)^{2+1/g}, while its counting function satisfies |A cap [1,x]| >> x^{g/(2g+1)}; Corollary 1.1 restates the representation part as a_3 <= n^{eps} for 0 < eps < 1 and every integer g > 1/eps, Corollary 1.2 records the counting part, and Corollary 1.3 is the case g = 2. The construction is probabilistic (a random set supported on the residue classes of a modular Sidon set from Theorem 3.1, Janson's inequality for the lower tail of the three-summand representation count, deletion of the largest summand of every (g+1)-fold representation, high moments for the upper tail of the deleted part, and Borel-Cantelli). The paper frames the whole line by the Erdős-Turán conjecture (Conjecture 1.1: no sequence and fixed g >= 2 has 1 <= r_A(n) <= g for all large n) and the stronger statement (1.4), which it attributes to Erdős and Fuchs, that every B_2[g] sequence with g >= 2 has liminf |A cap [1,x]|/x^{1/2} = 0; it records the case g = 1 as proved by Erdős with an extra factor (log x)^{1/2}. It also records Ruzsa's Sidon sequence with |A cap [1,x]| >> x^{sqrt(2)-1+o(1)} (and Cilleruelo's explicit construction of one). The general-g exponent g/(2g+1) improves the Erdős-Rényi bound, printed as x^{g/2(g+1)+o(1)}, and removes the factor (log x)^{-1/(2g+1)+o(1)} from Cilleruelo's bound (1.6) (p. 2, attributed on p. 3). At g = 2 it is 2/5, below Ruzsa's Sidon exponent sqrt(2) - 1, and a Sidon sequence is also B_2[2].

Read status: claims checked for every result listed under Results, read clause by clause on the page images of the print; the proof of Theorem 1.1 was read in outline only (Sections 3, 4, 5 and 10), and the moment estimates of Sections 6-9 were not checked. Nothing here is independently reviewed.

Source: https://arxiv.org/abs/2405.04154.

Bears on.

• #158: the problem's condition is the paper's B_2[2] (at most two representations a + b with a <= b). The case g = 2 of the conjectural statement (1.4) (p. 2) is the problem's question answered yes; the paper states it as a conjecture and proves nothing on it. Theorem 1.1 and Corollary 1.2 give, at g = 2, one infinite B_2[2] set with |A cap [1,x]| >> x^{2/5}; an exponent below 1/2 does not decide whether the liminf of the ratio to x^{1/2} is 0. The paper does not mention the problem.
• #28: Conjecture 1.1 (p. 1) is equivalent to the problem's statement, since the ordered count 1_A * 1_A(n) is bounded exactly when r_A(n) is; the paper states it as open and proves nothing on it.
• #157: Conjecture 1.2 (p. 4), for any one eps > 0, would give an infinite Sidon set that is an asymptotic basis of order 3, the object the problem asks for, with the further bound a_3 <= n^{eps}; the paper states it as a conjecture.

Contents.

• Conjecture 1.1 (p. 1): Erdős-Turán conjecture: no sequence A and fixed g >= 2 satisfy 1 <= r_A(n) <= g for all sufficiently large n.
• Theorem 1.1 (p. 2): For each g >= 2 there is a B_2[g] sequence A in which every large n is a_1+a_2+a_3 with a_3 << n^{1/g}(log n)^{2+1/g}, and its counting function satisfies |A cap [1,x]| >> x^{g/(2g+1)}.
• Corollary 1.1 (p. 2): For 0 < eps < 1 and any integer g > 1/eps there is a B_2[g] sequence in which every large n is a_1+a_2+a_3 with a_3 <= n^{eps}.
• Display (1.4) (p. 2): the conjectural statement, attributed to Erdős and Fuchs, that for any g >= 2 every B_2[g] sequence has liminf |A cap [1,x]|/x^{1/2} = 0.
• Display (1.5) (p. 2): Erdős's speculation that for every eps > 0 some Sidon sequence has |A cap [1,x]| >> x^{1/2-eps}, which the paper calls still well beyond reach, recalling Ruzsa's exponent sqrt(2)-1+o(1); this is not Conjecture 1.2.
• Corollary 1.2 (p. 3): For each integer g >= 2 there is a B_2[g] sequence A contained in N with |A cap [1,x]| >> x^{g/(2g+1)} for large x.
• Corollary 1.3 (p. 3): There is a B_2[2] sequence in which every large n is a_1+a_2+a_3 with a_3 << n^{1/2}(log n)^{5/2}. The paper calls this a stronger conclusion for g = 2 than Cilleruelo's proof that some B_2[2] sequence is an asymptotic basis of order 3, the existence Erdős and Nathanson claimed for some g.
• Conjecture 1.2 (p. 4): For each eps > 0, some Sidon set of positive integers represents every large n as a_1 + a_2 + a_3 with all three terms in the set and a_3 at most n^{eps}.
• Theorem 3.1 (p. 6; Cilleruelo, Proc. London Math. Soc. 111 (2015), Theorem 2.1, cited without proof): For infinitely many M, Z/MZ contains a Sidon set S such that every residue modulo M is s_1 + s_2 + s_3 with s_1, s_2, s_3 pairwise distinct elements of S.
• Proposition 4.1 (p. 8): Janson's inequality, cited from the literature, the concentration tool used to bound the lower tail of the representation count R_n(A) of the random sequence.

Results.

• Conjecture 1.1 (p. 1): the Erdős-Turán conjecture.
• Theorem 1.1 (p. 2): the main construction.
• Corollary 1.1 (p. 2): a_3 <= n^{eps} for g > 1/eps.
• Display (1.4) (p. 2): the conjectural liminf statement for B_2[g] sequences.
• Corollary 1.2 (p. 3): |A cap [1,x]| >> x^{g/(2g+1)}.
• Corollary 1.3 (p. 3): the case g = 2.
• Conjecture 1.2 (p. 4): the Sidon analogue of Corollary 1.1.

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