Ruzsa, Imre Z. Essential components. (English) Zbl 0609.10042 Proc. Lond. Math. Soc., III. Ser. 54, 38-56 (1987). 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 Cited in 2 ReviewsCited in 5 Documents 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 Keywords:sum-sets; sets of integers; asymptotic density; trigonometric sums; essential component PDF BibTeX XML Cite \textit{I. Z. Ruzsa}, Proc. Lond. Math. Soc. (3) 54, 38--56 (1987; Zbl 0609.10042) Full Text: DOI