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.


11B13 Additive bases, including sumsets


Zbl 0149.28503
Full Text: DOI Numdam


[1] Erdös, P., Számelméleti megjegyzések, III, Mat. Lapok13 (1962), 28-38. · Zbl 0123.25503
[2] Straus, E.G., On a problem in combinatorial number theory, J. Math. Sci.I (1966), 77-80. · Zbl 0149.28503
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