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Bulletin of the London Mathematical Society 2001 33(1):59-66; doi:10.1112/blms/33.1.59
© 2001 by London Mathematical Society
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© London Mathematical Society

How To Make Davies' Theorem Visible

Marianna Csörnyei

Department of Mathematics, University College London Gower Street, London WC1E 6BT; e-mail: mari{at}math.ucl.ac.uk

Received 10 June 1999. Revision received 18 December 1999.

We prove that for an arbitrary measurable set A sub R2 and a {sigma}-finite Borel measure µ on the plane, there is a Borel set of lines L such that for each point in A, the set of directions of those lines from L containing the point is a residual set, and, moreover, Formula We show how this result may be used to characterise the sets of the plane from which an invisible set is visible. We also characterise the rectifiable sets C1, C2 for which there is a set which is visible from C1 and invisible from C2.

Research supported by the Hungarian National Foundation for Scientific Research Grants F029768, T019476, and FKFP Grant 0192/1999.


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