Direct Sum Decomposition of the Integers
Yuji Ito · Tokyo Journal of Mathematics · 1995
In this paper we shall be concerned with the direct sum decomposition $Z=A\oplus B$ of the set $Z$ of all integers, where both subsets $A$ and $B$ are infinite subsets.Recently, a very interesting connection between such a decomposition of $Z$ (and of $N$ ) and properties of infinite measure preserving ergodic transformations was discovered, and exploiting this connection, a number of significant results have been obtained characterizing the nature of the summands that appear in such a decomposition, see [2], [3], [4] and [5].While it is well-known and is not difficult to characterize the infinite subsets that appear as direct summands of the decomposition $N=A\oplus B$ , see, for example [1], [6], the situation is very different for the case of the direct sum decomposition of $Z$ , where it seems to be very difficult to give a reasonable characterization of summands in general, see Proposition 2.2 below.On the other hand, if one fixes one of the summands of such a decomposition to be a "reasonable set" in some sense, then one can give some interesting characterizations for infinite subsets of $Z$ that can be a complement of this set in the decomposition of $Z$ .If we let the set $A$ to be one of the sets that appear as a direct summand of the decomposition of $N$ , for example, it follows from the known result mentioned above that there exists a unique subset $B$ such that $N=A\oplus B$ and it is not difficult to show that for this $B,$ $A\oplus(-B)=Z$ holds, and furthermore, one can construct by starting with this $B$ many other complements of $A$ in $Z$ , see Proposition 2.3 below.In fact, in [3] and [4], it was shown by using ergodic theory that such a set $A$ always has uncountably many distinct complements in $Z$ , some of which can be of very different nature from those described in Proposition 2.3.So, in this paper, we shall takeA to be one of the sets that can appear asadirect summand of the decomposition of $N$ ; in fact, for the sake of simplicity, we take $A$ to be the simplest of such sets, namely, let $A$ consist of $0$ and all finite sums of distinct odd powers of 2, and give a characterization of sets $C$ that can appear as a complement of this $A$ in the direct sum decomposition of $Z$ .