Everywhere of Second Category Sets

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.