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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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