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Journal of the London Mathematical Society Advance Access originally published online on March 16, 2009
Journal of the London Mathematical Society 2009 79(2):529-543; doi:10.1112/jlms/jdn086
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© 2009 London Mathematical Society

Large sets with limited tube occupancy

Anthony Carbery

School of Mathematics and Maxwell Institute for Mathematical Sciences
University of Edinburgh
JCMB, King's Buildings
Mayfield Road
Edinburgh, EH9 3JZ
Scotland

We study subsets E of euclidean space with the property that for every tube, the amount of mass of E contained in that tube is small, and address via the probabilistic method the question of how large such sets may be. We also study discrete analogues of this question, and relate it to problems in harmonic analysis concerning the extension operator for the Fourier transform.

Part of this research was undertaken at the University of Athens in spring 2006, where the author was supported by a Leverhulme Study Abroad Fellowship and the EC project ‘Pythagoras’.

2000 Mathematics Subject Classification 28A78, 42B99, 52C99, 60D05.

Received August 22, 2008; published online March 16, 2009.


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