## 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 G1 and G2 of R both of which are subspaces of R over the field of rational such that G1\cap G2 = {0},G1+G2 = R and both G1 and G2 have Lebesgue measure zero. Theorem 12. Assume 2\aleph0 = \aleph1. 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

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