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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
Bulletin of the London Mathematical Society – Wiley
Published: Sep 1, 1986
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