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Statement

Setting (p. 2, Definition 1). A sequence $A$ of positive integers is a $B_2[g]$ sequence when every integer $n$ has at most $g$ representations $n=a+a'$ with $a\le a'$ and $a,a'\in A$; the $B_2[1]$ sequences are the Sidon sequences. $A$ is an asymptotic basis of order $h$ when every sufficiently large positive integer is a sum of $h$ elements of $A$ (p. 1).

Theorem 1.2 (p. 2, quoted). "There exists a $B_2[2]$ sequence of positive integers which is an asymptotic basis of order $3$."

The paper reports (p. 2) that Erdős claimed some $B_2[g]$ sequence is an asymptotic basis of order $3$ and asked for the least such $g$; Conjecture 1.1 would give $g=1$, and this theorem gives $g\le2$. It also records (p. 3) that the standard probabilistic argument already gives a $B_2[3]$ sequence that is an asymptotic basis of order three, citing Alon and Spencer's book (§8.6).

Source. J. Cilleruelo, On Sidon sets and asymptotic bases, Proceedings of the London Mathematical Society 111 (2015), 1206--1230, read in arXiv:1304.5351v2 (titled "Sidon basis") as identified on the source card; labels and pages are that preprint's.

Read depth. Claims checked: the statement, Definition 1 and the strategy of Section 4.1 were read clause by clause on the page images. The proof and the expected-value computations of Section 6 were not checked step by step. Nothing here is independently reviewed.

Proof pointer

Section 4 (pp. 14--17), with expected values in Section 6.1 (pp. 22--25). Fix $\mathbb{Z}_N$ and a Sidon set $S\subset\mathbb{Z}_N$ from Theorem 2.1 (p. 4), every element a sum of three pairwise distinct elements of $S$. The random sequence lives in the space $\mathcal{S}_m(7/11;S\bmod N)$ of Definition 3 (p. 11): the events $x\in A$ are independent, with $\mathbb{P}(x\in A)=x^{-7/11}$ when $x>m$ and $x$ lies in a residue class of $S$ modulo $N$, and $0$ otherwise; the paper notes that any $\gamma$ with $5/8<\gamma<2/3$ would do (p. 14). The $B_2[2]$-lifting process of Definition 6 (p. 14) deletes each element that is a summand of a sum with three different representations, and the survivors form a $B_2[2]$ sequence (p. 15). Janson's inequality and Borel-Cantelli give, with probability 1, at least a constant times $n^{1/11}$ representations of each large $n$ as a sum of three elements in distinct classes (Proposition 4.1, p. 15), and a vectorial sunflower argument shows that, with probability $1-o_m(1)$, at most $10^{28}$ of them are destroyed for every $n$ (Proposition 4.2, p. 16).

Bears on

• Problem 157: the problem asks for a Sidon set, that is a $B_2[1]$ sequence, that is an asymptotic basis of order 3. This theorem gives such a basis with the Sidon condition relaxed to $B_2[2]$; it does not settle the problem.
• Problem 158: the problem's sets, infinite with at most two solutions of $a+b=n$ with $a\le b$, are the infinite $B_2[2]$ sequences of Definition 1. The theorem constructs one that is an asymptotic basis of order 3, but the paper states no bound on its counting function at the scale $N^{1/2}$, and the theorem does not bear on the liminf the problem asks about.