Title: EVERYWHERE OF SECOND CATEGORY SETS
Authors: Kandasamy Muthuvel and Arnold W. Miller
Abstract:
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.
LaTeX 2.09 8 pages. Postcript and TeX file available at
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Real Analysis Exchange, 24(1999), 607-614.