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The Realization of small Distances in Plane Sets of Positive Measure

The Realization of small Distances in Plane Sets of Positive Measure THE REALIZATION OF SMALL DISTANCES IN PLANE SETS OF POSITIVE MEASURE K. J. FALCONER Consider the set of real numbers E = {xe U: 2n ^ x < 2n+\ for some integer n). It is trivial to verify that E has asymptotic measure density \, and that the distance set of E (that is the set [\x— y\: x,yeE}) does not contain any odd integers. Thus for each positive integer n the set 3~ E (pointwise multiplication) has density \ but does 1 2 n not contain the distances 1,3" ,3~ ,...,3" . A naive attempt to construct 'large' plane sets avoiding such distance sequences might be to note that the set F = E x E cz U does not contain the distance 1 (though it does contain all sufficiently large distances), so that the set -1 2 n 1 2 n G = Fn4 Fn 4~ F0 ... n4- Fdoe s not contain the distances l,^ ,^ , ...,4~ . (n+1) However, the plane density of G is 16~ which tends to zero exponentially with n. The purpose of this note is to show that this is quite typical; the plane situation contrasting sharply with the 1-dimensional http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the London Mathematical Society Wiley

The Realization of small Distances in Plane Sets of Positive Measure

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Publisher
Wiley
Copyright
© London Mathematical Society
ISSN
0024-6093
eISSN
1469-2120
DOI
10.1112/blms/18.5.475
Publisher site
See Article on Publisher Site

Abstract

THE REALIZATION OF SMALL DISTANCES IN PLANE SETS OF POSITIVE MEASURE K. J. FALCONER Consider the set of real numbers E = {xe U: 2n ^ x < 2n+\ for some integer n). It is trivial to verify that E has asymptotic measure density \, and that the distance set of E (that is the set [\x— y\: x,yeE}) does not contain any odd integers. Thus for each positive integer n the set 3~ E (pointwise multiplication) has density \ but does 1 2 n not contain the distances 1,3" ,3~ ,...,3" . A naive attempt to construct 'large' plane sets avoiding such distance sequences might be to note that the set F = E x E cz U does not contain the distance 1 (though it does contain all sufficiently large distances), so that the set -1 2 n 1 2 n G = Fn4 Fn 4~ F0 ... n4- Fdoe s not contain the distances l,^ ,^ , ...,4~ . (n+1) However, the plane density of G is 16~ which tends to zero exponentially with n. The purpose of this note is to show that this is quite typical; the plane situation contrasting sharply with the 1-dimensional

Journal

Bulletin of the London Mathematical SocietyWiley

Published: Sep 1, 1986

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