Perelli, A.; Zannier, U. On sums of distinct integers belonging to certain sequences. (English) Zbl 0519.10050 Acta Math. Hung. 41, 251-254 (1983). The present paper deals with the classical problem of additive number theory of representing integers as sums of distinct terms taken from a fixed sequence. Here combinatorial methods are used to prove the following result: Let \(\mathcal A\) be a sequence of natural numbers such that \(1\in\mathcal A\), and for every large \(x\), \((x, 2x) \cap\mathcal A \ne \emptyset\). Then there exists a number \(L=L(\mathcal A)\) with the following property: if \(\mathcal B\) is a sequence of natural numbers such that (i) if \(a, a' \in \mathcal A\), \(b,b' \in\mathcal B\) and \(ab = a'b'\) then \(a = a'\) and \(b = b'\); (ii) \(\vert \{b\in\mathcal B, b\le y\}\vert > L \log^2y\) for some \(y > 10\), then \(S = \{m, m = a_1+\cdots+ a_r,\ a_1 > \cdots > a_r,\ a_j\in \mathcal{AB}\}\) (here we use \(\mathcal{AB}= \{ab, a\in \mathcal A, b\in \mathcal B\})\) contains an arithmetical progression. From this result one obtains that, if the G.C.D. of all large terms of \(\mathcal A\) is \(1\), then \(S\) contains all large integers. The above theorem is perhaps not optimal: in particular cases one obtains stronger results [cf. B. J. Birch, Proc. Camb. Philos. Soc. 55, 370–373 (1959; Zbl 0093.05003)], but it covers situations which escape from theorems proved by sophisticated analytical methods [cf. J. W. S. Cassels, Acta Sci. Math. 21, 111–124 (1960; Zbl 0217.32102)]. Reviewer: Alberto Perelli (Genova) Page: −5 −4 −3 −2 −1 ±0 +1 +2 +3 +4 +5 Show Scanned Page Cited in 1 Document MSC: 11B13 Additive bases, including sumsets 11B83 Special sequences and polynomials 11B25 Arithmetic progressions Keywords:sums of distinct integers; arithmetical progression Citations:Zbl 0093.05003; Zbl 0217.32102 PDFBibTeX XMLCite \textit{A. Perelli} and \textit{U. Zannier}, Acta Math. Hung. 41, 251--254 (1983; Zbl 0519.10050) Full Text: DOI References: [1] B. J. Birch, Note on a problem of Erdös,Proc. Cambridge Phil. Soc.,55 (1959), 370–373. · Zbl 0093.05003 · doi:10.1017/S0305004100034150 [2] J. W. S. Cassels, On the representation of integers as the sums of distinct summands taken from a fixed set,Acta Sci. Math. Szeged,21 (1960), 111–124. · Zbl 0217.32102 [3] P. Erdös, On the representation of large integers as sums of distinct summands taken from a fixed set,Acta Arith.,7 (1960), 345–354. · Zbl 0106.03805 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.