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On sums of distinct integers belonging to certain sequences. (English) Zbl 0519.10050

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)].

MSC:

11B13 Additive bases, including sumsets
11B83 Special sequences and polynomials
11B25 Arithmetic progressions
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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
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