The Mathematics of Paul Erdős I pp 533-540 | Cite as

# On Lipschitz Mappings Onto a Square

Chapter

First Online:

## Abstract

The following problem was posed by Laczkovich [5]: Let \(E \subseteq \mathbb{R}{\mathbb{R}}^{d}\) (*d* ≥ 2) be a set with positive Lebesgue measure *λ* ^{ d }(*E*) > 0. Does there exist a Lipschitz mapping \(f : {\mathbb{R}}^{d} \rightarrow Q = {[0,1]}^{d}\), such that *f*(*E*) = *Q*? Preiss [6] answered this question affirmatively for *d* = 2:

## Notes

### Acknowledgements

I would like to thank David Preiss for explaining me his proof, useful discussions and for his hospitality during my visit.

### References

- 1.G. Alberti, M. Csörnyei, D. Preiss: Structure of null sets in the plane and applications, in
*European Congress of Mathematics: Stockholm, June 27–July 2, 2004 (A. Laptev ed.)*, Europ. Math. Soc., Zurich 2005, pages 3–22.Google Scholar - 2.G. Alberti, M. Csörnyei, D. Preiss: Differentiability of Lipschitz functions, structure of null sets, and other problems, in
*Proc. ICM 2010*, vol. III, World Scientific, Hackensack, NJ, pages 1379–1394.Google Scholar - 3.N. Alon, J. Spencer, P. Erdös:
*The probabilistic method.*Cambridge Univ. Press 1992.MATHGoogle Scholar - 4.P. Erdös, G. Szekerés: A combinatorial problem in geometry.
*Compositio Math.*2(1935) 463–470.MathSciNetGoogle Scholar - 5.M. Laczkovich: Paradoxical decompositions using Lipschitz functions,
*Real Analysis Exchange*17(1991–92), 439–443.Google Scholar - 6.D. Preiss, manuscript, 1992.Google Scholar
- 7.T. Szabó, G. Tardos: A multidimensional generalization of the Erdős–Szekeres lemma on monotone subsequences,
*Combinatorics, Probability and Computing*10(2001) 557–565.Google Scholar - 8.N. X. Uy, Removable sets of analytic functions satisfying a Lipschitz condition,
*Ark. Mat.*17(1979), 19–27.MathSciNetMATHCrossRefGoogle Scholar - 9.J. H. Wells, L. R. Williams:
*Embeddings and extensions in analysis*, Springer-Verlag 1975.Google Scholar

## Copyright information

© Springer Science+Business Media New York 2013