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Statement

Setting (p. 1). For $g\in\mathbb N$, $B_2[g]$ is the class of sets $A\subset\mathbb N$ such that for every $n\in\mathbb N$ the equation $a+a'=n$ with $a,a'\in A$ and $a\le a'$ has at most $g$ solutions; $B_2[1]$ is the class of Sidon sets. The counting function is $A(x)=\#\{a\le x : a\in A\}$.

Theorem 1 (p. 2, quoted). "For all $g\ge2$ there exists an infinite $B_2[g]$ sequence $A$ such that $\limsup_{x\to\infty}\frac{A(x)}{\sqrt x}=L_g$ where"

$$L_g= \begin{cases} \sqrt{3/2}, & g=2,\\ 3/2, & g=3,\\ \sqrt{36/11}, & g=4,\\ \sqrt{9/2}, & g=5,\\ \sqrt{100/17}, & g=6,\\ \sqrt{27/4}, & g=7,\\ \sqrt8, & g=8,\\ \dfrac{3}{2\sqrt2}\sqrt{g-1}, & g\ge9. \end{cases}$$

In particular (the case $g=2$) there is an infinite $B_2[2]$ sequence with $\limsup_{x\to\infty}A(x)/\sqrt x=\sqrt{3/2}$, against the value $1$ of Kolountzakis's earlier infinite $B_2[2]$ sequence that the introduction cites (p. 1).

The abstract (p. 1) states the formula $\frac{3}{2\sqrt2}\sqrt{g-1}$ for every $g\ge2$, with better values for small $g$. Compared here with the table: the tabulated value equals the formula at $g=3,5,7$, exceeds it at $g=2,6,8$, and falls below it at $g=4$ ($\sqrt{36/11}\approx1.809$ against $\sqrt{27/8}\approx1.837$).

Scope of the printed proof

For $g\ge9$ the proof of Proposition 2 (p. 4) takes $C_g=A^{g-1}$, where $A^g={k : 0\le k\le g-1}\cup{g-1+2k : 1\le k\le [g/2]}$ is a set the paper credits to Cilleruelo, Ruzsa and Trujillo (reference [1], a preprint), and asserts that property iii), the ratio $|C_g|/\sqrt{u_g+1}=L_g$, is easy to see. Computed here: with $u_g$ the largest element of $C_g$, the ratio is $\frac{3}{2\sqrt2}\sqrt{g-1}$ when $g$ is odd, but $\frac{3g-4}{2\sqrt{2g-3}}$ when $g$ is even, slightly smaller (at $g=10$, $13/\sqrt{17}\approx3.153$ against $\approx3.182$); a larger $u_g$ only lowers it. So in the version read the proof reaches the stated $L_g$ for $2\le g\le8$ and for odd $g\ge9$, and for even $g\ge10$ it gives an infinite $B_2[g]$ sequence with the smaller value $\frac{3g-4}{2\sqrt{2g-3}}$ in place of $L_g$.

Proof pointer

Pp. 2--4. Any finite $B_2[g]$ set $A_0$ with largest element $x$ is extended by a block $D=\bigcup_{c\in C_g}(B_p+cm+2x)$, where $p$ is a prime with $x^2<p<2x^2$ and $m=p^2-1$. Proposition 1 (p. 2) supplies $B_p\subset(p^{1/2},p^2-p^{1/2})$ with more than $p-4p^{1/2}$ elements, pairwise more than $p^{1/2}$ apart, whose pairwise sums are distinct modulo $m$; it is cut down (p. 4) from a modular Sidon set of $p$ elements in $[1,p^2-1]$ that the paper attributes to Chowla and Erdős, citing Halberstam and Roth. Proposition 2 (p. 2) supplies an integer $u_g$ and a set $C_g\subset[0,u_g]$ whose representation function $r$ satisfies i) $r(n)\le g$ for all $n$, ii) $r(c)\le g-1$ and $r(c-1)\le g-1$ for $c\in C_g$, and iii) $|C_g|/\sqrt{u_g+1}=L_g$; for $g\le8$ the sets are listed explicitly on p. 4. Proposition 3 (p. 3) shows that $A_0\cup D$ is again $B_2[g]$, by reducing a representation to one of the form $c+c'$ (or $c+c'$ equal to $c_0$ or $c_0-1$ when one summand lies in $A_0$), and Proposition 4 (p. 4) shows that the ratio at the last element is $L_g+o(1)$. Repeating the extension gives the infinite sequence.

Read depth

Claims checked: the definition of $B_2[g]$ and of $A(x)$, the statement of Theorem 1 with its table, and the statements of Propositions 1 to 4 were read clause by clause on the pages of the print; the proofs were read but not checked step by step. Two checks were computed here: the arithmetic of the table and of $|C_g|/\sqrt{u_g+1}$ above, and properties i) and ii) of Proposition 2 for the listed sets $C_2,\ldots,C_8$ and for $A^{g-1}$ with $9\le g\le20$, which hold whether $r$ counts ordered or unordered pairs (the paper does not say which; the reduction in Proposition 3 needs ordered pairs). Nothing here is independently reviewed.

Dependencies

None in the corpus. External inputs named by the paper: the Chowla--Erdős modular Sidon set (via Halberstam and Roth, Sequences) and the representation bound for $A^g$ from the Cilleruelo--Ruzsa--Trujillo preprint.

Source. J. Cilleruelo and C. Trujillo, Infinite $B_2[g]$ sequences, Israel Journal of Mathematics 126 (2001), 263--267, doi:10.1007/BF02784156, read in the four-page author-typeset version named on the source card; pages here are that version's printed pages 1--4, not the journal's.

Bears on

• Problem 158: the $g=2$ case gives an infinite set of the problem's kind (at most two solutions of $a+b=n$ with $a\le b$) with $\limsup A(N)/N^{1/2}=\sqrt{3/2}$. The problem asks about the limit inferior, on which the theorem says nothing; it neither answers nor refutes the question.
• Problem 329: the problem asks for the largest $\limsup A(N)/N^{1/2}$ of a Sidon set ($g=1$). The theorem concerns $g\ge2$, a wider class of sets, and gives no bound for Sidon sets; the problem's site lists the paper among its references.