On the center of distances
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Abstract
We introduce the notion of a center of distances of a metric space and use it in a generalization of the theorem by John von Neumann on permutations of two sequences with the same set of cluster points in a compact metric space. This notion is also used to study sets of subsums of some sequences of positive reals, as well for some impossibility proofs. We compute the center of distances of the Cantorval, which is the set of subsums of the sequence \(\frac{3}{4}, \frac{1}{2}, \frac{3}{16}, \frac{1}{8}, \ldots , \frac{3}{4^n}, \frac{2}{4^n}, \ldots \), and for other related subsets of the reals.
Keywords
Cantorval Center of distances von Neumann’s theorem Set of subsums Digital representationMathematics Subject Classification
40A05 11B05 28A751 Introduction
The center of distances seems to be an elementary and natural notion which, as far as we know, has not been studied in the literature. It is an intuitive and natural concept which allows us to prove a generalization of von Neumann’s theorem on permutations of two sequences with the same set of cluster points in a compact metric space, see Theorem 2.1. We have realized that the computation of centers of distances—even for wellknown metric spaces—is not an easy task because it requires skillful use of fractions. We have only found a few algorithms which enable us to compute centers of distances, see Proposition 3.2 and Lemma 5.1.
We present the use of this notion for impossibility proofs, i.e., to show that a given set cannot be the set of subsums, for example see Corollary 5.5. We refer the readers to the paper [14] by Nitecki, as it provides a good introduction to facts about the set of subsums of a given sequence. It is also worth to look into the papers [1, 2, 3, 8] as well as others cited therein.
2 A generalization of von Neumann’s theorem
Given a metric space X with the distance d. Suppose that sequences \(( x_n)_{n\in \omega } \) and \((y_n)_{ n\in \omega } \) in X have the same set of cluster points C. For them, von Neumann [13] proved that there exists a permutation \(\pi :\omega \rightarrow \omega \) such that Open image in new window Proofs of the above statement can be found in [6, 18]. However, we would like to present a slight generalization of this result. To prove it we use the socalled “backandforth” method, which was developed in [7, pp. 35–36] and is still used successfully by many mathematicians, for example cf. [4, 16] or [17], etc. It is also worth mentioning the modern development of classical works of Fraïssé by Kubiś [11].
Theorem 2.1
Suppose that sequences Open image in new window and Open image in new window in X have the same set of cluster points \( C \subseteq X\), where (X, d) is a compact metric space. If \(\alpha \in S(C)\), then there exists a permutation \(\pi :\omega \rightarrow \omega \) such that Open image in new window
Proof
If Open image in new window is not defined, then take points \( x_m, y_m \in C\) such that Open image in new window and \(d(x_m, y_m)=\alpha \). Choose Open image in new window to be the first element of Open image in new window not already used such that Open image in new window
If Open image in new window is not defined, then take points \( p_m, q_m \in C\) such that Open image in new window and \(d(p_m, q_m)=\alpha \). Choose Open image in new window to be the first element of Open image in new window not already used such that Open image in new window
Let us note that von Neumann’s theorem mentioned above is applicable for some other problems, for example cf. [9] or [10], etc. As we have seen, the notion of a center of distances appears in a natural way in the context of metric spaces. Though the computation of centers of distances is not an easy task, it can be done for important examples giving further information about these objects.
3 On the center of distances and the set of subsums
Given a metric space X, observe that \(0\in S(X)\) and also, if Open image in new window and \(0\in X\), then \(S(X) \subseteq X\).
Proposition 3.1
If X is the set of subsums of a sequence Open image in new window , then \(a_n\in S(X)\), for all \(n\in \omega \).
Proof
Suppose Open image in new window If \(n\in A\), then \(xa_n\in X\) and \(d(x, xa_n)= a_n\). When \(n\notin A\), then \(x+a_n\in X\) and \(d(x, x+a_n)= a_n\). \(\square \)
In some cases, the center of distances of the set of subsums of a given sequence can be determined. For example, the unit interval is the set of subsums of the sequence Open image in new window So, the center of distances of the subsums of Open image in new window is equal to Open image in new window
Proposition 3.2
Assume that Open image in new window , for a number \(\lambda >0\) and a set Open image in new window If Open image in new window and \(n \in \omega \), then \(\lambda ^n x\notin X\).
Proof
Using Proposition 3.2 with \(\lambda = \frac{1}{q^n} \) and \(b=1\), one can prove the next theorem. In fact, this proposition explains the hidden argument in the next proof.
Theorem 3.3
If \(q>2\) and \(a> 0\), then the center of distances of the set of subsums of a geometric sequence Open image in new window consists of exactly zero and the terms of this sequence.
Proof
Now, assume that \(n>0\) is fixed. Suppose that Open image in new window and Open image in new window Thus Open image in new window witnesses that \(t \notin S(X)\). Indeed, Open image in new window implies Open image in new window , and Open image in new window implies that the Xgap Open image in new window has to contain Open image in new window
By Proposition 3.1, we get \(\frac{1}{q^n} \in S(X)\) for \(n>0\). \(\square \)
Note that, when we put \(a = 2\) and \(q = 3\), Theorem 3.3 applies to the Cantor ternary set. For Open image in new window and \(q=4\) this theorem applies to sets \(\mathscr {C}_1\) and \(\mathscr {C}_2\) which will be defined in Sect. 4.
4 An example of a Cantorval
Proposition 4.1
Reals from \(K_n\) are distributed consecutively at the distance \(\frac{1}{4^n}\), from Open image in new window up to Open image in new window , in the Open image in new window Therefore Open image in new window and Open image in new window
Proof
Corollary 4.2
The interval Open image in new window is included in the Cantorval \( \mathbb {X}\).
Proof
The union Open image in new window is dense in the interval Open image in new window \(\square \)
Note that it has been observed that Open image in new window , see [5] or cf. [14]. Since \(\mathbb {X}\) is centrally symmetric with \(\frac{5}{6}\) as a point of inversion, this yields another proof of the above corollary. However, our proof seems to be new and it is different from the one included in [5].
Put Open image in new window , for \(n\in \omega \). So, each \(C_n\) is an affine copy of \(\mathbb {X}\).
Proposition 4.3
The subset Open image in new window is the union of pairwise disjoint affine copies of \(\mathbb {X}\). In particular, this union includes two isometric copies of Open image in new window , for every \(n>0\).
Proof
The desired affine copies of \(\mathbb {X}\) are \(C_1\) and Open image in new window , \(\frac{1}{2} + C_2\) and Open image in new window , and so on, i.e., Open image in new window and Open image in new window \(\square \)
Proposition 4.4
The subset Open image in new window is the union of six pairwise disjoint affine copies of Open image in new window
Proof
The desired affine copies of Open image in new window lie as shown in Fig. 2. \(\square \)
Corollary 4.5
The Cantorval Open image in new window has Lebesgue measure 1.
Proof
There exists a onetoone correspondence between \(\mathbb {X}\)gaps and \(\mathbb {X}\)intervals as it is shown in Fig. 3.
5 Computing centers of distances
In case of subsets of the real line we formulate the following lemma.
Lemma 5.1
Given a set Open image in new window disjoint from an interval Open image in new window , assume that Open image in new window Then the center of distances S(C) is disjoint from the interval Open image in new window , i.e., Open image in new window
Proof
We will apply the above lemma by putting suitable Cgaps in the place of the interval Open image in new window In order to obtain \(t\notin S(C)\), we must find \(x <t\) such that Open image in new window and \(x\in C\). For example, this is possible when \(\frac{\alpha }{2}< t < \alpha \) and the interval Open image in new window includes no Cgap of the length greater than or equal to \(\beta \alpha \). But if such a gap exists, then we choose the required x more carefully.
Theorem 5.2
The center of distances of the Cantorval \(\mathbb {X}\) is equal to Open image in new window
Proof
The diameter of \(\mathbb {X}\) is \(\frac{5}{3}\) and \(\frac{5}{6} \in \mathbb {X}\), hence no \(t>\frac{5}{6}\) belongs to \(S(\mathbb {X})\). We use Lemma 5.1 with respect to the gap Open image in new window Keeping in mind the affine description of \(\mathbb {X}\), we see that the set Open image in new window has a gap Open image in new window of the length \(\frac{1}{12}\). For Open image in new window , we choose x in \(\mathbb {X}\) such that Open image in new window So, if Open image in new window , then \(t\notin S(\mathbb {X})\). Similarly using Lemma 5.1 with the gap Open image in new window , we check that for Open image in new window there exists x in \(\mathbb {X}\) such that Open image in new window Hence, if Open image in new window , then \(t\notin S(\mathbb {X})\). Analogously, using Lemma 5.1 with the gap Open image in new window , we check that if \(\frac{5}{24}< t \leqslant \frac{29}{96} < \frac{5}{12}\), then \(t\notin S(\mathbb {X})\).
For the remaining part of the interval \([0,+\infty )\) the proof uses the similarity of \(\mathbb {X}\) with \(\frac{1}{4^n}{\cdot } \mathbb {X}\) for \(n>0\). Indeed, we have shown that the \(\mathbb {X}\)gaps Open image in new window and Open image in new window witness that Open image in new window For \(n>0\), by the similarity, the \(\mathbb {X}\)gaps Open image in new window , Open image in new window and Open image in new window witness that Open image in new window
We have by Proposition 3.1. \(\square \)
Denote Open image in new window Thus the closure of an \(\mathbb {X}\)gap is a \(\mathbb {Z}\)interval and the interior of an \(\mathbb {X}\)interval is a \(\mathbb {Z}\)gap.
Theorem 5.3
The center of distances of the set \(\mathbb {Z}\) is trivial, i.e., \(S(\mathbb {Z})= \{0\}\).
Proof
Now, denote Open image in new window Thus, each \(\mathbb {X}\)gap is also a \(\mathbb {Y}\)gap, and the interior of an \(\mathbb {X}\)interval is a \(\mathbb {Y}\)gap.
Theorem 5.4
Proof
Corollary 5.5
Neither \(\mathbb {Z}\) nor \(\mathbb {Y}\) is the set of subsums of a sequence.
Proof
Since \(S(\mathbb {Z})=\{0\}\), Proposition 3.1 decides the case of \(\mathbb {Z}\). Also, this proposition decides the cases of \(\mathbb {Y}\), since Open image in new window \(\square \)
Let us add that the set of subsums of the sequence Open image in new window is included in \(\mathbb {Y}.\) One can check this, observing that each number Open image in new window , where the nonempty set \(A\subset \omega \) is finite, is the right end of an \(\mathbb {X}\)interval.
6 Digital representation of points in the Cantorval \(\mathbb {X}\)
Suppose \(n_1\) is the least index such that \(a_{n_1}\!=5\) and \(b_{n_1}\!=0\), thus B is chasing A in the \(n_1\)step. Proceeding this way, we obtain an increasing (finite or infinite) sequence \(n_0< n_1 < \cdots \) such that Open image in new window and Open image in new window for Open image in new window Moreover, A starts chasing B in the \(n_k\)step for even k’s and B starts chasing A in the \(n_k\)step for odd k’s, for the rest of steps changes of chasing do not occur.
Theorem 6.1

Open image in new window , whenever \(0<k<n_0\);

\(a_{n_0}\!= 2\) and \(b_{n_0}\!=3\);

\(a_{n_k}\!= 5\) and \(b_{n_k}\!=0\), for odd k;

\(a_{n_k}\!= 0\) and \(b_{n_k}\!=5\), for even \(k>0\);

\(a_i \in \{3,5\}\) and \(a_i  b_i =3\), whenever Open image in new window

\(a_i \in \{0,2\}\) and \(b_i  a_i =3\), whenever Open image in new window
Proof
According to the chasing algorithm described above in the step \(n_k \) the roles of chasing are reversed. But, if the chasing algorithm does not start, then the considered point has a unique digital representation. \(\square \)
The above theorem makes it easy to check the uniqueness of digital representation. For example, if \(x \in \mathbb {X}\) has a digital representation \((x_n)_{n>0}\) such that \(x_n=2\) and \(x_{n+1}=3\) for infinitely many n, then this representation is unique. Indeed, suppose Open image in new window and Open image in new window are two different digital representations of a point \(x\in \mathbb {X}\) such that \(a_k =b_k\), whenever \(0<k<n_0\) and \(a_{n_0}\!= 2 < b_{n_0}\!=3\). By Theorem 6.1, the digit 3 never occurs immediately after the digit 2 in digital representations of x for the digits greater than \(n_0\), since it has to be Open image in new window for \(k>n_0\).
Theorem 6.2
Let \(A\subset B\) be such that Open image in new window and A are infinite. Then the set of subsums of a sequence consisting of different elements of A is homeomorphic to the Cantor set.
Proof
Fix a nonempty open interval \(\mathbb {I}\). Assume that Open image in new window is the digital representation of a point Open image in new window Choose natural numbers \(m>k\) such that numbers Open image in new window and Open image in new window belong to \(\mathbb {I}\). Then choose \(j>m\) such that Open image in new window , where \(a =2\) or \(a=3\). Finally put \(b_{n} = a_{n}\), for \( 0<n \leqslant k\); Open image in new window and \(b_{i} = 0\) for other cases. Since \(b_m=0\), we get Open image in new window Theorem 6.1 together with conditions Open image in new window and Open image in new window imply that the point Open image in new window is not in the set of subsums of A. Thus, this set being dense in itself and closed is homeomorphic to the Cantor set. \(\square \)
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