APPLICATION OF TRANSLATION OF SETS

Kandasamy Muthuvel

University of Wisconsin–Oshkosh

In this paper, we show that if a set A of reals contains a translated copy of every finite set then for any (additive) proper subgroup F of R, A \ F contains a translated copy of every finite set. This implies that the real line is not the finite union of proper subgroups of R. It is interesting to note that no group is the union of two proper subgroups but there is a group, namely {1, 3, 5, 7} under addition modulo 8, which is the union of three proper subgroups. Existence of small sets of reals (in the sense of category and measure) containing a translated copy of every countable set is proved in [3].

Throughout this paper, R denotes the set of all real numbers, N is the set of all positive integers and R^{∗} denotes the set of all nonzero reals.

\underline{Proposition 1}. If F is a proper subgroup of R, then |R : F |, the index of F in R is infinite.

\underline{Proof}. Suppose |R : F | is finite. Then R is a finite union of left cosets F +x_{i}, 1 ≤ i ≤ m. Since {nx_{i} : n ∈ N} is infinite for each i, for infinitely many n, nx_{i} belongs to the same coset, say F + x_{j}. Hence there exists a smallest positive integer Y_{i} such that Y_{i}x_{i} ∈ F, because n_{1}x_{i} and n_{2}x_{i} ∈ F + x_{j} for some n_{1}, n_{2} imply that (n_{1} − n_{2})x_{i} ∈ F. Let ℓ be the least common multiple of Y_{i} for 1 ≤ i ≤ m. Then ℓx_{i} ∈ F for every i ≤ m. For any r ∈ R, r = f + x_{i} for some f ∈ F and some i ≤ m. Hence ℓr ∈ F for all r ∈ R, and consequently ℓ(\frac{r}{ℓ} ) = r ∈ F for all r ∈ R, which contradicts that F is a proper subgroup of R.

\underline{Corollary 1}. R is not the direct sum of a cyclic subgroup and a proper subgroup of R. \underline{Proof}. If R =< g > +F, then \frac{g}{2} = ng + f for some n ∈ Z and f ∈ F. Then

(2n − 1)g ∈ F and consequently R is a finite union of left cosets of F in G, contradicting Proposition 1.

\underline{Remark 1}. It can be easily seen from the proof of Proposition 1 that if G is any additive subgroup of R such that \frac{g}{n} ∈ G for all g ∈ G and n ∈ N or G is the multiplicative subgroup of R^{∗} such that g^{\frac{1}{n}} ∈ G for all g ∈ G and n ∈ N, then G contains no proper subgroup of finite index. For example, the set of all rationals under addition or the set of all positive real numbers under multiplication contains no proper subgroup of finite index.

\underline{Proposition 2}. If a subset of A of R contains a translated copy of every finite set, then for any proper subgroup F of R, A \ F contains a translated copy of every finite set.

\underline{Proof}. Let H be a finite set of n elements. Since |R : F | is infinite, R ⊇ F + r_{i} for every i ∈ N, where r_{i} − r_{j} ∉ F whenever i ≠ j and i, j ∈ N. Since H + {r_{i} : 1 ≤ i ≤ n + 1} is finite, H + {r_{i} : 1 ≤ i ≤ n + 1} + r ⊆ A for some r ∈ R. It suffices to prove that H + r_{i} + r ∩ F = ∅ for some i ≤ n + 1. Suppose H + r_{i} + r ∩ F ≠ ∅ for every i ≤ n + 1. Then for each i ≤ n + 1, there exists h_{i} ∈ H such that h_{i} + r_{i} + r ∈ F. Since |H| = n and there are n + 1 values for i, h_{j} = h_{ℓ} for some j ≠ ℓ and 1 ≤ j, ℓ ≤ n + 1. Hence

h_{j} + r_{j} + r − (h_{ℓ} + r_{ℓ} + r) = r_{j} − r_{ℓ} ∈ F ,

which contradicts the fact r_{j} − r_{ℓ} ∉ F.

\underline{Corollary 2}. R is not the finite union of proper subgroups of R.

\underline{Proof}. Suppose

R = ⋃ F_{i} ,

where F_{i} is a proper subgroup of R and 1 ≤ i ≤ n. Then by Proposition 2, R \ F_{1} contains a translated copy of every finite set. By continuing in this fashion,

R \ ⋃_{1≤i≤n} F_{i} = ∅

contains a translated copy of every finite set. This is impossible.

\underline{Definition}. The algebraic difference of a set A of reals, denoted by D(A), is defined to be {x − y : x, y ∈ A}.

A classical result of Piccard [4] states that if A is a Baire set and is of second category, then D(A) = A − A contains an interval.

The following lemma (stated without proof in [2]) is slightly stronger than Piccard’s theorem.

\underline{Lemma 1}. If Q is a set of the first Baire category and a < b, then

D((a, b) \ Q) = D(a, b) = (a − b, b − a).

First we shall prove a lemma, which is slightly stronger than the above lemma, to prove a proposition.

\underline{Lemma 2}. Let (a, b) be an interval and x a fixed element in (a, b). If a set A contains a translated copy of {x, y} for every y ∈ (a, b), then D(A) ⊇ (a − x, b − x).

\underline{Proof}. If not, find an element i in (a − x, b − x) \ D(A). Then \{x, x + i\} + r ⊆ A for some r in R, and consequently i ∈ A − A = D(A), which contradicts the choice of i.

\underline{Proposition 3}. Let a < b. If Q is a set of the first Baire category, or Q is a subgroup of R of uncountable index, then D((a, b) \ Q) = (a − b, b − a).

\underline{Proof}. Let x be a fixed element in (a, b). Suppose Q is a set of the first Baire category. Then for every y in (a, b), {x, y} + r ⊆ (a, b) for all r ∈ (0, b − max{x, y}) = I (say). If {x, y} + r ∩ Q ≠ ∅ for every r ∈ I, then there exists a second category subset J of I such that for all r ∈ J, x + r ∈ Q or y + r ∈ Q for all r ∈ J. This contradicts that Q is of first category. Thus (a, b)\Q satisfies the hypothesis of Lemma 2. Now, suppose Q is a subgroup of R of uncountable index. If {x, y} + r ∩ Q ≠ ∅ for all r ∈ I, then I ⊆ (Q − x) ∪ (Q − y). By using the fact that any open interval of length |I| is a translated copy of I, it can be easily seen that R is contained in a countable union of sets of the form Q − x, which contradicts that |R : Q| is uncountable. Thus (a, b) \ Q contains a translated copy of {x, y} for every y ∈ (a, b).

In both cases, by Lemma 2, D((a, b) \ Q) ⊇ (a − x, b − x) for every x ∈ (a, b) and consequently D((a, b) \ Q) ⊇ (a − b, b − a). Trivially D((a, b) \ Q) ⊆ (a − b, b − a). This completes the proof.

\underline{References}

1. J. A. Gallian, Contemporary Abstract Algebra, D. C. Heath and Company, Lexington, Massachusetts, 1990.

2. H. I. Miller and L. Miller, “A Result about Porous Sets and Difference Sets,” Real Analysis Exchange, 13 (1987–88), 441–445.

3. P. Komjath, “Large small sets,” Coll. Math, 56 (1988), 231–233.

4. J. C. Oxtoby, Measure and Category, Springer–Verlag, 1970, New York.