# Some additive properties of sets of real numbers

@article{Erds1981SomeAP, title={Some additive properties of sets of real numbers}, author={Paul Erd{\"o}s and Kenneth Kunen and R. Daniel Mauldin}, journal={Fundamenta Mathematicae}, year={1981}, volume={113}, pages={187-199} }

Some problems concerning the additive properties of subsets of R are investigated. From a result of G. G . Lorentz in additive number theory, we show that if P is a nonempty perfect subset of R, then there is a perfect set M with Lebesgue measure zero so that P+M = R. In contrast to this, it is shown that (1) if S is a subset of R is concentrated about a countable set C, then A(S+R) = 0, for every closed set P with A(P) = 0 ; (2) there are subsets G, and G s of R both of which are subspaces of…

## 55 Citations

Some properties of Borel subgroups of real numbers

- Mathematics
- 1983

As a consequence of Souslin's theorem, we obtain the following: if G and II both are analytic subgroups of R such that G + I = R and G n II {O}= then either G = R or G = (0). Next we obtain some…

SOME ADDITIVE PROPERTIES OF SPECIAL SETS OF REALS

- Mathematics
- 2007

D. H. Fremlin and J. Jasiński [4] have proved a relative consistency of the existence of a very thin set of reals. In this context they have asked (private communication) the following question:…

Nonmeasurable algebraic sums of sets of reals

- Mathematics
- 2005

We present a theorem which generalizes some known theorems on the existence of nonmeasurable (in various senses) sets of the form X+Y . Some additional related questions concerning measure, category…

Homogeneity for open partitions of pairs of reals

- Mathematics
- 1993

We prove a partition theorem for analytic sets of reals, namely, if A ⊆ R is analytic and [A] 2 = K 0 ∪ K 1 with K 0 relatively open, then either there is a perfect 0-homogeneous subset or A is a…

ADDITIVE PROPERTIES OF SETS OF REAL NUMBERS AND AN INFINITE GAME

- Mathematics
- 1993

Abstract If X has strong measure zero aid if Y is contained in an F σ, set of measure zero, then X + Y has measure zero (Proposition 9). If X is a measure zero set with property s 0 and Y is a…

The Algebraic Sum of Sets of Real Numbers with Strong Measure Zero Sets

- MathematicsJ. Symb. Log.
- 1998

These results extend: Fremlin and Miller's theorem that strong measure zero sets having Hurewicz's property have Rothberger's property, Galvin and Miller’s theorem that the algebraic sum of a set with the γ-property and of a first category set is aFirst category set, and Bartoszyfnski and Judah's characterization of -sets.

COMPLEXITY OF INDEX SETS OF DESCRIPTIVE SET-THEORETIC NOTIONS

- Mathematics, Computer Science
- 2018

A generalization of computability theory, admissible recursion theory, is applied to consider the relative complexity of notions that are of interest in descriptive set theory, and it is demonstrated that there is a separation of descriptive complexity between the perfect set property and determinacy for analytic sets of reals.

## References

SHOWING 1-6 OF 6 REFERENCES

On a Problem of Additive Number Theory

- Mathematics
- 1956

where all the prime factors of each A,are of a given form. A search of the literature seemed to indicate that various theorems had been conjectured but none actually proved.f For example, L. Euler…

Ensembles à coupes dénombrables et capacités dominées par une mesure

- Mathematics
- 1978

© Springer-Verlag, Berlin Heidelberg New York, 1978, tous droits réservés. L’accès aux archives du séminaire de probabilités (Strasbourg) (http://portail. mathdoc.fr/SemProba/) implique l’accord avec…

Sommes vectorielles d'ensembles de mesure nulle

- Art, Mathematics
- 1975

© Annales de l’institut Fourier, 1976, tous droits réservés. L’accès aux archives de la revue « Annales de l’institut Fourier » (http://annalif.ujf-grenoble.fr/) implique l’accord avec les conditions…

Un ensemble singular

- Bull . Soc . Math
- 1980