Hostname: page-component-78c5997874-m6dg7 Total loading time: 0 Render date: 2024-11-06T09:52:26.699Z Has data issue: false hasContentIssue false

Subfields of R with arbitrary Hausdorff dimension

Published online by Cambridge University Press:  31 March 2016

R. DANIEL MAULDIN*
Affiliation:
5383 Renaissance Avenue, San Diego, CA 92122, U.S.A. e-mail: [email protected]

Abstract

Assuming CH, the continuum hypothesis, holds we show, by completing an attack first discovered by Roy Davies, that for each α between 0 and 1 there is a subring, in fact a subfield, of R with Hausdorff dimension α.

Type
Research Article
Copyright
Copyright © Cambridge Philosophical Society 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1]Bourgain, J.On the Erdös–Volkmann and Katz–Tao ring conjectures. Geom. Funct. Anal. 13 (2003), no. 2, 334365. MR 1982147 (2004d:11070)Google Scholar
[2]Bugeaud, Y.Intersective sets and Diophantine approximation. Michigan Math. J. 52 (2004), 667682.Google Scholar
[3]Croft, H. T., Falconer, K. J. and Guy, R. K.Unsolved Problems in Geometry (Springer-Verlag, New York, 1991).Google Scholar
[4]Edgar, G. A. and Miller, C.Borel subrings of the reals. Proc. Amer. Math. Soc. 131 (2002), 11211129.CrossRefGoogle Scholar
[5]Erdös, P. and Volkmann, B.Additive Gruppen mit vorgegebener Hausdorffscher dimension. J. Reine Angew. Math. 221 (1966), 203208.Google Scholar
[6]Falconer, K.Rings of fractional dimension. Mathematika 31 (1984), 2527.Google Scholar
[7]Falconer, K.On the Hausdorff dimension of distance sets. Mathematika 32 (1985), 206212.Google Scholar
[8]Falconer, K.Sets with large intersection properties. J. Lond. Math. Soc. 49 (1994), 267280.Google Scholar
[9]Falconer, K.Fractal geometry, second ed. (John Wiley and Sons Inc., Hoboken, NJ, 2003), Mathematical foundations and applications. MR MR2118797 (2006b:28001).Google Scholar
[10]Jackson, S. and Mauldin, R. D.On a lattice problem of H. Steinhaus. J. Amer. Math. Soc. 15 (2002), no. 4, 817856.Google Scholar
[11]Mattila, P.Geometry of Sets and Measures in Euclidean Spaces (Cambridge University Press, 1995).Google Scholar
[12]Rynne, B.Regular and ubiquitous systems and ${\mathcal M}^s_\infty$-dense sequences. Mathematika 39 (1992), 234243.Google Scholar
[13]Volkmann, B.Eine metrische Eigenschaft reeller Zählkörper. Math. Ann. 141 (1960), 237238.Google Scholar