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**Zentralblatt MATH**

**Publications of (and about) Paul Erdös**

**Zbl.No: ** 209.35402

**Autor: ** Erdös, Paul; Rényi, Alfréd

**Title: ** On some applications of probability methods to additive number theoretic problems (In English)

**Source: ** Contrib. Ergodic Theory Probab., Lecture Notes Math. 160, 37-44 (1970).

**Review: ** [For the entire collection see Zbl 202.05002.]

The authors prove that to every \alpha, 0 < \alpha < 1 there is a sequence A of density \alpha so that for very sequence of integers b_{1} < ... < b_{k} the density of \bigcup^{k}_{i = 1} **{**A+b_{i}**}** is 1-(1-\alpha)^{k}. The theorem no longer holds if 1-(1-\alpha)^{k} is replaced by 1-(1-\alpha)^{k}+\epsilon. But the authors believe that there is a sequence of density \alpha so that the density of \bigcup^{k}_{i = 1} **{**A+b_{i}**}** is always greater than 1-(1-\alpha)^{k}. The authors also answer a question of A. Stöhr by showing that there is a sequence A of density 0, so that for every basis B A+B has density 1. The methods of the proof are probabilistic.

**Classif.: ** * 11B05 Topology etc. of sets of numbers

11K99 Probabilistic theory

**Citations: ** Zbl 202.050(EA)

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