Open Access
1998/1999 Everywhere of Second Category Sets
Arnold W. Miller, Kandasamy Muthuvel
Real Anal. Exchange 24(2): 607-614 (1998/1999).


The main result of this paper states the following. For each natural number i, let $G_i$ be a proper additive subgroup of the reals, $A_i$ a set that contains no arithmetic progression of length three, $H_i$ a basis for the vector space $\R$ over the field of rationals, and $E^{+}(H_i)$ the set of all finite linear combinations from the elements of $H_i$ with nonnegative rational coefficients. Then the complement of a finite union of sets $G_i\cup A_i\cup E^{+}(H_i)$ is everywhere of second category. We also prove that the complement of a union of fewer than continuum many translates of sets that have distinct distances is everywhere of second category.


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Arnold W. Miller. Kandasamy Muthuvel. "Everywhere of Second Category Sets." Real Anal. Exchange 24 (2) 607 - 614, 1998/1999.


Published: 1998/1999
First available in Project Euclid: 28 September 2010

zbMATH: 0968.26002
MathSciNet: MR1704737

Primary: 26A03

Keywords: arithmetic progression , everywhere of second category , set having distinct distances

Rights: Copyright © 1999 Michigan State University Press

Vol.24 • No. 2 • 1998/1999
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