Application of covering sets

Abstract

This paper contains results concerning covering sets which generalize and unify some known results about the additive subgroups of the reals and the algebraic difference of sets.Throughout the paper, the set of all real numbers is denoted by R. The algebraic difference of a subset A of R is defined to be A − A = {x − y : x, y ∈ A}. Any basis for the vector space of the reals over the rationals is called a Hamel basis.Sierpiński proved that the complement of a Hamel basis is everywhere of the second category. This was generalized in [5]: if H is a Hamel basis and X is a subset of R of cardinality less than the cardinality of the continuum, then the complement of the algebraic sum H +X is everywhere of the second category.The above result was improved in [6] by showing that (1) the complement of the algebraic sum Z(H) + X is everywhere of the second category, where |X| < |R|, and the Erdős set, Z(H), is the set of all finite linear combinations of elements from a Hamel basis H with integer coefficients, and (2) the complement of a finite union of Hamel bases is everywhere of the second category. Z(H) + X is contained in a proper subgroup of the additive group of the reals because the cardinality of the group generated by X is less than the cardinality of the continuum and the index of the additive subgroup Z(H) of R is the cardinality of the continuum.In this paper, we generalize and unify the above results by showing that the complement of a finite union of proper subgroups of R is everywhere of the second category (i.e., it is large in the sense of category). We prove a theorem about covering sets which directly implies that the cardinality of the complement of a finite union of proper subgroups of R is the cardinality of the continuum. We also prove a theorem which generalizes and unifies Theorems 5, 6, and 8 of [3].Notation. We use the standard set theory notation. The set of all real numbers and the set of all natural numbers are denoted by R and N re-1991 Mathematics Subject Classification: Primary 26A03; Secondary 04A15, 54A25.[115]

On the Steinhaus property in topological groups

“…Since, as observed at the end of Section 3.2, every Erdős subgroup of R has index c in R and is neither a Lebesgue null nor a first category set, we directly obtain from Theorem 4.7 that a nonempty union of fewer than c-many translates of finitely many Erdős subgroups of R is λ-saturated and lacks the Baire property everywhere. This observation sharpens Corollary 4 of [16] (cf. also [15,Theorem 6.III.16]).…”

“…Combine Theorems 3.4 and 4.1. 2 Theorem 4.6 extends in various respects Muthuvel's Theorem 2 in[16] stating that the complement of a finite union of proper subgroups of R is everywhere of second category. Theorem 4.7 below establishes the mensural counterpart of Theorem 4.6.…”

“…The problem of "estimating" Z c gains a particular interest in a topological setting, namely when G is also supposed to be a Hausdorff topological group. In the case where G = R and x 1 = · · · = x n = 0, Muthuvel [16,Theorem 2] proves that Z c is everywhere of second category. We give in Theorem 4.6 a substantial generalization of his result, in particular replacing R with an arbitrary connected Baire group.…”

Dichotomy and infinite combinatorics: the theorems of Steinhaus and Ostrowski

“…Extensions of Neumann's Lemma for abelian groups are considered in [22] and also by Muthuvel in [65], [66], [67]. The fact that a finite number of proper cosets cannot cover the reals follows from a number of theorems including those of Laczkovich [54] and Miller and Muthuvel [63].…”

Bernstein sets with algebraic properties

“…Coverings have been studied extensively in the case of κ = ω (see [6] and [7]). The following proposition is folklore, but we include it for completeness.…”