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Probabilistic constructions in additive number theory. (English) Zbl 0625.10046
Journées arithmétiques, Besançon/France 1985, Astérisque 147/148, 173-182 (1986).
[For the entire collection see Zbl 0605.00004.]
The author uses a new, probabilistic approach for constructing a class of sets \(H\subseteq {\mathbb{Z}}\) with the property that, for any set \(A\subseteq {\mathbb{Z}}\) whose counting function A(x) is not bounded by any function g(x) with g(x)/log x \(\nearrow\) and g(x)/x \(\searrow 0\), both \(H+A\) and H-A have (asymptotically) many elements in a sense specified for either of the two cases.
The result yields solutions of several older problems in additive number theory including the existence of a sequence A of density 0 such that the Schnirelman density \(\sigma\) satisfies \(\sigma (A+B)>0\) for every base B.
Reviewer: B.Volkmann

11B13 Additive bases, including sumsets
11A25 Arithmetic functions; related numbers; inversion formulas
11K99 Probabilistic theory: distribution modulo \(1\); metric theory of algorithms
11B83 Special sequences and polynomials