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Essential components. (English) Zbl 0609.10042
The author calls a set H of nonnegative integers an essential component if ḏ(A\(+H)>\underline d(A)\) for any set A with ḏ(A)\(<1\). (Here ḏ denotes lower asymptotic, not Schnirelman density!). It is proved by probabilistic methods that there exist, for every \(\epsilon >0\), essential components satisfying \(H(x)=O(\log^{1+\epsilon} x)\). Furthermore, it is shown that, for any essential component H, there exist numbers \(c>0\) and \(x_ 0\) such that \(H(x)>\log^{1+\epsilon} x\), \(\forall x>x_ 0\). One of the main tools is a characterization of essential components in terms of additive behavior modulo m \((m=1,2,...)\).
Reviewer: B.Volkmann

MSC:
11B05 Density, gaps, topology
11B13 Additive bases, including sumsets
11K99 Probabilistic theory: distribution modulo \(1\); metric theory of algorithms
11P55 Applications of the Hardy-Littlewood method
11B83 Special sequences and polynomials
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