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Source. Alain Plagne, Recent progress on finite $B_h[g]$ sets, author's manuscript (no venue or year printed), Section 2.2 (pp. 4-8), the material used here on pp. 5-7, as identified on the source card. The file prints no page numbers; pages are counted from its first page.

Statement

Setting (p. 1, formula (1)). For integers $h\ge2$ and $g\ge1$, a set $\mathcal A$ of integers is $B_h[g]$ when every integer $n$ has at most $g$ representations $n=a_1+\cdots+a_h$ with $a_i\in\mathcal A$ and $a_1\le\cdots\le a_h$. $F_{h,g}(N)$ is the largest size of a $B_h[g]$ subset of $\{1,\ldots,N\}$, and $f(N)\lesssim g(N)$ means $f(N)\le(1+o(1))g(N)$ as $N\to\infty$ (p. 2).

Ordered variant (p. 6). $\mathcal A$ is $B_h^*[g]$ when every integer $n$ has at most $g$ ordered representations $(a_1,\ldots,a_h)\in\mathcal A^h$ with $a_1+\cdots+a_h=n$. The paper notes that a $B_h[g]$ set is $B_h^*[gh!]$.

Gluing principle (p. 5, formula (10)). If $\{a_0=0,\ldots,a_k\}$ is a $B_h[g]$ set and $C$ is a $B_h[1]$ set modulo $m$, then $\bigcup_{i=0}^k(C+ma_i)$ is a $B_h[gh!]$ set of integers, which gives $F_{h,gh!}(N)/N^{1/h}\gtrsim(k+1)/(1+a_k)^{1/h}$.

Definition and inequality (11) (p. 6). Let $\mathcal M_{h,g}$ be the set of numbers $(k+1)/(1+a_k)^{1/h}$ over all $k\ge0$ and all $B_h^*[g]$ sets $\{a_0,\ldots,a_k\}$, and $\mu_{h,g}=\sup\mathcal M_{h,g}$. The paper states that $x\in\mathcal M_{h,g}$ implies $F_{h,g}(N)N^{-1/h}\gtrsim x$, hence

$$\frac{F_{h,g}(N)}{N^{1/h}}\gtrsim\mu_{h,g}.\qquad(11)$$

Problem 2 (p. 6, quoted). "Given $g$ and $h$, show that $\mu_{h,g}$ is attained, identify the set which reaches this supremum and find the value of $\mu_{h,g}$ (or at least its asymptotic behavior when $g$ and $h$ are large)."

The paper says it is natural to conjecture that the supremum is attained by a relatively small set (p. 6).

The case $h=g=2$ (pp. 6-7). The paper reports that Habsieger and the author answered Problem 2 for $B_2[2]$ in its reference [15] (L. Habsieger, A. Plagne, Ensembles $B_2[2]$ : l'étau se resserre, submitted 2000): $\mu_{2,2}$ is attained by the Sidon set $\{0,1,4,6\}$, so $\mu_{2,2}=4/\sqrt7$, and

$$F_{2,2}(N)\gtrsim\frac4{\sqrt7}\sqrt N,\qquad(12)$$

with $4/\sqrt7=1.5118\ldots$, against the $3/2$ that formula (6) (Cilleruelo, Ruzsa and Trujillo) gives.

Read depth. Claims checked: the definitions, inequality (11), the problem and the report of the case $h=g=2$ were read clause by clause on pp. 1-2 and 5-7. The proof of (12) is in [15] and was not read here.

Proof pointer

The paper gives no proof of (11) beyond the gluing principle (10), applied to $B_h^*[g]$ seeds (p. 6). The answer for $h=g=2$ is cited to [15].

Dependencies

The gluing construction of p. 5, which needs modular $B_h[1]$ sets such as those of Bose and Chowla (p. 4).

Bears on

No Erdős problem page states this question. The bound (12) is a lower bound on the finite function $F_{2,2}(N)$; see the source card for its relation to Problem 158.