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Source. Display (1.4), p. 2, of Javier Pliego, On the Erdős-Turán conjecture and the growth of $B_2[g]$ sequences, arXiv preprint arXiv:2405.04154v1 (7 May 2024), the version named on the source card. The statement is unnumbered apart from its display label.

Statement

Setting (p. 1): $r_A(m)$ counts unordered pairs $\{a_1,a_2\}\subset A$ with $a_1+a_2=m$, a sum $a+a$ counting once, and $A\subset\mathbb N$ is $B_2[g]$ when $r_A(m)\le g$ for every $m\in\mathbb N$.

The paper calls this "the stronger conjectural statement (see Erdős and Fuchs [11])" (p. 2, quoted; [11] is Erdős and Fuchs, On a problem of additive number theory, J. London Math. Soc. 31 (1956)): for any $g\ge2$, every $B_2[g]$ sequence $A\subset\mathbb N$ satisfies

$$\liminf_{x\to\infty}\frac{\lvert A\cap[1,x]\rvert}{x^{1/2}}=0. \tag{1.4}$$

The paper observes that it implies Conjecture 1.1, since an asymptotic basis of order 2 cannot satisfy (1.4). It records the case $g=1$ as proved by Erdős, citing Halberstam and Roth, Sequences, §2 Theorem 8: every Sidon sequence has $\liminf_{x\to\infty}\lvert A\cap[1,x]\rvert(\log x)^{1/2}/x^{1/2}=0$.

Scope

A conjecture the paper records and does not attack; its Theorem 1.1 gives a lower bound $x^{g/(2g+1)}$ for one $B_2[g]$ sequence, below $x^{1/2}$ and so consistent with (1.4).

Read depth. Claims checked: the sentence, the display and the attribution were read clause by clause on the page image of p. 2. The attribution to Erdős and Fuchs is the paper's; the cited 1956 paper was not compared here.

Bears on

• Problem 158: the case $g=2$ of (1.4) is the problem's question answered yes, in the same convention (at most two representations $a+b$ with $a\le b$); for a finite set the lower limit is $0$ trivially, so only infinite sets matter. The paper states it as a conjecture and proves nothing about it.