---

Statement

A set $\mathcal A\subset\mathbb N$ is admissible when, in the words of the English abstract (printed p. 55), "the sums of the elements of two subsets of $\mathcal A$ of different cardinalities are different"; equivalently, writing $\mathcal P(\mathcal A,k)$ for the set of integers that are sums of exactly $k$ distinct elements of $\mathcal A$ and $P(\mathcal A,k)$ for its size, $k\ne\ell$ implies $\mathcal P(\mathcal A,k)\cap\mathcal P(\mathcal A,\ell)=\emptyset$ (printed p. 55). $F(N)$ is the maximum cardinality of an admissible subset of $\mathbb N_N=\{1,\ldots,N\}$ (p. 56).

Théorème 1 (printed p. 56, display (3)).

$$\limsup_{N\to\infty}F(N)\,N^{-1/2}\le(143/27)^{1/2}\quad(=2.301368\ldots).$$

The paper adds that a more elaborate application of its ideas could improve the constant, but that reaching the conjectured limit $\limsup F(N)N^{-1/2}=2$ in this way seems impossible and that a new idea seems necessary for any upper bound below $2.2$ (p. 56).

The introduction (p. 56) reports Straus's results, from his 1966 paper (not held here): (i)(1) $\limsup F(N)N^{-1/2}\le4/\sqrt3$ $(=2.309401\ldots)$; Erdős's conjecture that $F(N)$ is attained by a set of consecutive integers including $N$, $\mathcal A=\{N-F(N)+i:1\le i\le F(N)\}$; and (ii) the set $\{N-k+1,\ldots,N\}$ is admissible for $k=2m-1$ if $m^2\le N<m^2+m$ and for $k=2m$ if $m^2+m\le N<(m+1)^2$, which implies (2) $\liminf F(N)N^{-1/2}\ge2$.

Source. P. Erdős, J.-L. Nicolas and A. Sárközy, Sommes de sous-ensembles, Sém. Théor. Nombres Bordeaux (2) 3 (1991), no. 1, 55–72 (Journal de théorie des nombres de Bordeaux; DOI 10.5802/jtnb.42; the Numdam record read); the copy read for this page is the Numdam file of the article, 19 pages, printed p. $n$ on PDF p. $n-53$. Théorème 1 and the Straus passage on printed p. 56 (PDF p. 3), read on the page image; the French is rendered here in the corpus's words, the displays as printed.

Read depth. Claims checked: the definition, Théorème 1 and the account of Straus's results were read clause by clause on the page image. The proof (Section 3, pp. 58–62) was read for its structure only.

Proof pointer

Section 3 (printed pp. 58–62) distinguishes three cases for an admissible $\mathcal A=\{a_1<\cdots<a_y\}\subseteq\mathbb N_N$, according to how many elements lie in $]17N/18,N]$ (the set $\mathcal A_3$) and in $[1,N/2]$ (the set $\mathcal A_1$): $|\mathcal A_3|<\frac{21}{32}y$ (display (6)); $|\mathcal A_3|\ge\frac{21}{32}y$ and $|\mathcal A_1|<\frac1{32}y$ ((8) and (9)); $|\mathcal A_3|\ge\frac{21}{32}y$ and $|\mathcal A_1|\ge\frac1{32}y$ ((13) and (14)); and in each bounds $y$ through counts of $P(\mathcal A,k)$ from Lemme 1 ($P(\mathcal A,k)\ge k(|\mathcal A|-k)+1$, Straus's Theorem 2) and the disjointness of the sets $\mathcal P(\mathcal A,k)$ inside $[1,kN]$; the three case bounds (7), (12) and (19) give the theorem. Not reconstructed here.

Dependencies

Straus's Theorem 2 (Lemme 1) and Theorem 4 (Lemme 2), quoted from E. G. Straus, J. Math. Sci. 1 (1966), 77–80, not held; the paper proves Lemme 2 from Lemme 1 and states Lemme 1 with a one-line proof pointer.

Bears on

• Problem 874: the intermediate upper bound the site quotes, $\limsup k(N)/N^{1/2}\le(143/27)^{1/2}=2.301\cdots$, and the paper's own record of Straus's $4/\sqrt3=2.309\cdots$, of the block construction giving $\liminf\ge2$, and of Erdős's conjecture, proved for large $N$ by Deshouillers and Freiman (Theorem 1). The paper's $F(N)$ is the problem's $k(N)$.