Direct Sum Decomposition of the Integers

Abstract

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