##
**Zentralblatt MATH**

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

**Zbl.No: ** 482.28001

**Autor: ** Erdös, Paul; Kunen, K.; Mauldin, R.Daniel

**Title: ** Some additive properties of sets of real numbers. (In English)

**Source: ** Fundam. Math. 113, 187-199 (1981).

**Review: ** The paper is concerned with some additive properties of subsets of the real line R. The following finite version of *G.G.Lorentz*'s theorem [Proc. Am. Math. Soc. 5, 838-841 (1954; Zbl 056.039)] is proved: There is a positive number c so that for any positive integers n, m, and k, if A is a set of integers, A\subset[m,m+k], with |A| \geq l, there is a set B of integers, B\subset[n,n+2k] so that A+B: = **{**a+b: a in A,b in B**}** contains all integers in the interval (n+m+k,n+m+2k] with |B| < c log l/l. The following theorems are also obtained: Theorem 4. If S is a subset of R which is concentrated about a countable subset C, Then \lambda(S+P) = 0, for every closed set P with Lebesgue measure zero. Theorem 5. There are subsets G_{1} and G_{2} of R both of which are subspaces of R over the field of rational such that G_{1}\cap G_{2} = **{**0**}**,G_{1}+G_{2} = R and both G_{1} and G_{2} have Lebesgue measure zero. Theorem 12. Assume 2^{\aleph0} = \aleph_{1}. Then there is a subset X of R such that (1) |X| = \aleph, (2) \forall G\subseteq R[\lambda(G) = 0 ==> \lambda(G+X) = 0], (3) X is concentrated on the rationals. Open questions: Can one prove in ZFC that there is an X satisfying (1) and (2) of theorem 12?

**Reviewer: ** M.S.Marinov

**Classif.: ** * 28A05 Classes of sets

03E15 Descriptive set theory (logic)

28A12 Measures and their generalizations

**Keywords: ** ZFC; Lorentz's theorem

**Citations: ** Zbl.056.039

© European Mathematical Society & FIZ Karlsruhe & Springer-Verlag