## Subset sums. (Sommes de sous-ensembles.)(French)Zbl 0742.11008

For any set $${\mathfrak A}\subseteq\mathbb{N}$$ let $${\mathfrak P}({\mathfrak A},k)$$ be the set of all $$n$$ which are expressible as sums of exactly $$k$$ distinct elements of $${\mathfrak A}$$. The set $${\mathfrak A}$$ is called admissible if, for $$k\neq\ell$$, $${\mathfrak P}({\mathfrak A},k)\cap{\mathfrak P}({\mathfrak A},\ell)=0$$. The authors prove that, if $$F(N)$$ is the maximal cardinality of admissible sets $${\mathfrak A}\subseteq\{1,\ldots,N\}$$, then $$\limsup_{N\to\infty}F(N)N^{-1/2}\to(143/27)^{1/2}$$, thereby slightly sharpening a result due to E. G. Straus [J. Math. Sci. 1, 77–80 (1966; Zbl 0149.28503)]. The paper also contains a result on infinite admissible sets and some general conjectures.