Complementing sets of integers-A result from ergodic theory

Abstract

Weakly wandering sequences play an important role in the study of ergodic measure preserving transformations defined on an infinite measure space. A se quence {ni} is a weakly wandering sequence for a transformation T defined on a measure space (X, B, Vie) if there exists a set A of positive measure for which the sets Tni A are mutually disjoint. In the past we have been successful in utilizing these sequences in exhibiting new and useful properties of ergodic transformations. In this article we discuss an application of weakly wandering sequences to the additive behavior of integers. We present here a result and a proof that are number theoretic in nature. We should mention however, that the measure preserving transforma tions that we have been studying together with their ergodic properties are the underlying concrete structures from which we were able to conjecture and prove the algebraic results that follow. In this article we omit discussion of topics that are ergodic in nature. To the interested reader we refer the articles [2] through [6]. Let us consider the set of all non-negative integers N={0, 1, 2,...}. Two infinite subsets A and B of N are said to have a direct sum if whenever an integer nEN can be represented as n=a+b and n=a'+b' with a, a'EA and b, b'EB then a=a' and b=b'. In this case we use the notation A+B to indicate the set {a+baEA, beB} . If A+BN then we say that B is a complementing set for A in N. Clearly, this is a symmetric property in A and B. De Bruijn [1] studied the behavior of complementing subsets of N, and the structure of such subsets is well understood, see [1] and [7]. When the set N is replaced by the set Z, the set of all integers, then the situation gets to be much more complicated, and very little seems to be known about the behavior of complementing subsets of Z. For instance, if an infinite set A has a complementing set B in N then this complementing set in N is unique. It is also not difficult to show that the set A has the set -B={-bbEB} as a complementing set in Z, and in fact, the set A happens to possess a continuum number of complementing subsets in Z; see [3]. A typical example in this connection is the set A consisting of 0 and all finite sums of odd powers of 2. One may regard the set A as consisting of 0 and all positive integers n which contain zero in the even places of their binary representation.It is easy to see that A has as its complementing set in N the set B consisting

Universally bad integers and the 2-adics

“…Some examples of good pairs include (1,1), (1,7), and (7,13). A list of some good pairs ða; bÞ can be found in [3] (this 1964 list was made with the aid of a computer, and it extends his earlier 1950 list in [2] for 1papbp100 obtained by ''pencil and paper'' and ''shuffling four strips of paper''-these quotes are from de Bruijn's 1964 paper [3]; see our acknowledgement at the end of our paper).…”

Direct Sum Decomposition of the Integers

“…For $n\in Z$ , we define $ord_{2}(n)$ to be equal to $k$ if $2^{k}$ divides $n$ but $2^{k+1}$ does not. In [5], the following result was obtained: THEOREM 3.1 (Eigen, Hajian and Kakutani). If $F$ is a finite subset of $Z$ , then there exists a $C\in \mathfrak{C}(A)$ such that $C\supset F$ if and only if every number belonging to the difference set $F-F$ is of even $ord_{2}$ .…”

“…

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

…”

“…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].…”

“…S. Kakutani gave in [5] a very interesting characterization for a finite subset $F$ of $Z$ in order for it to be extendable to a complement of $A$ in $Z$ . We will prove in \S 3 a theorem related to this result giving a necessary and sufficient condition for an infinite subset $C$ to be a complement of $A$ in $Z$ .…”