Real Analysis Exchange Vol. 24(2), 1998/9, pp. 615–619

Krzysztof Ciesielski^{∗}, Department of Mathematics, West Virginia University, Morgantown, WV 26506-6310, USA. e-mail: KCies@wvnvms.wvnet.edu Saharon Shelah^{†}, Institute of Mathematics, the Hebrew University of Jerusalem, 91904 Jerusalem, Israel, and Department of Mathematics, Rutgers University, New Brunswick, NJ 08854, USA

UNIFORMLY ANTISYMMETRIC FUNCTIONS WITH BOUNDED RANGE

Abstract

The goal of this note is to construct a uniformly antisymmetric function f: ℝ → ℝ with a bounded countable range. This answers Problem 1(b) of Ciesielski and Larson [6]. (See also the list of problems in Thomson [9] and Problem 2(b) from Ciesielski’s survey [5].) A problem of existence of uniformly antisymmetric function f: ℝ → ℝ with finite range remains open.

A function f: ℝ → ℝ is said to be uniformly antisymmetric [6] (or nowhere weakly symmetrically continuous [9]) provided for every x ∈ ℝ the limit lim_{n→∞}(f(x + s_{n}) − f(x − s_{n})) equals 0 for no sequence {s_{n}}_{n<ω} converging to 0. Uniformly antisymmetric functions have been studied by Kostyrko [7], Ciesielski and Larson [6], Komjáth and Shelah [8], and Ciesielski [1, 2]. (A connection of some of these results to the paradoxical decompositions of the Euclidean space ℝ^{n} is described in Ciesielski [3].) In particular in [6] the authors constructed a uniformly antisymmetric function f: ℝ → ℕ and noticed that the existence of a uniformly antisymmetric function cannot be proved without an essential use of the axiom of choice.

Key Words: uniformly antisymmetric function, Hamel basis

Mathematical Reviews subject classification: 26A15, 26A21

Received by the editors May 3, 1998

∗The first author was partially supported by NSF Cooperative Research Grant INT-9600548, with its Polish part being financed by Polish Academy of Science PAN, and 1996/97 West Virginia University Senate Research Grant.

Papers authored or co-authored by a Contributing Editor are managed by a Managing Editor or one of the other Contributing Editors.

The authors like to thank the referee for many valuable comments and suggestions.

†This work was supported in part by a grant from “Basic Research Foundation” of the Israel Academy of Sciences and Humanities. Publication 680.

The terminology and notation used in this note is standard and follows [4]. In particular for a set X we will write |X| for its cardinality and P(X) for its power set. Also 2^{ω} will stand for the set of all functions from ω = {0, 1, 2, … } into 2 = {0, 1}. We consider 2^{ω} as ordered lexicographically.

Theorem 1. There exists a function f: ℝ → ℝ with countable bounded range such that for every x ∈ ℝ there exists an ε_{x} > 0 with the property that the set

S_{x} = {s ∈ ℝ: |f(x − s) − f(x + s)| < ε_{x}}

is finite. In particular f is uniformly antisymmetric.

Proof. First notice that it is enough to find a compact zerodimensional metric space ⟨T, d⟩ and a function g from ℝ into a countable subset T_{0} of T such that for every x ∈ ℝ there is a δ_{x} > 0 for which the set

Ŝ_{x} = {s ∈ ℝ: d(g(x − s), g(x + s)) < δ_{x}}

is finite.

To see this assume that such a function g: ℝ → T exists and take a homeomorphic embedding h of T into ℝ. We claim that f = h ∘ g: ℝ → ℝ is as desired. Indeed, f[ℝ] = h[g[ℝ]] is countable, as it is a subset of a countable set h[T_{0}], and it is bounded, since it is a subset of a compact set h[T]. So take x ∈ ℝ and δ_{x} > 0 for which Ŝ_{x} is finite. Since h^{−1} : h[T] → T is uniformly continuous, we can find an ε_{x} > 0 such that

|y_{1} − y_{2}| < ε_{x} implies d(h^{−1}(y_{1}), h^{−1}(y_{1})) < δ_{x}

for every y_{1}, y_{2} ∈ h[T]. But for such a choice of ε_{x} we have

S_{x} = {s ∈ ℝ: |h(g(x − s)) − h(g(x + s))| < ε_{x}} ⊂ Ŝ_{x}

proving that S_{x} is finite.

Thus, we proceed to construct a function g described above. The value of g(x) will be defined with help of a representation of x in a Hamel basis; i.e., a linear basis of ℝ over ℚ. For this we will use the following notation. Let {y_{η}: η ∈ 2^{ω}} be a one-to-one enumeration of a Hamel basis H. For every x ∈ ℝ let ∑_{η∈2^{ω}} q_{x,η}y_{η}, with q_{x,η} ∈ ℚ for η ∈ 2^{ω}, be the unique representation of x in basis H and let w_{x} = {η ∈ 2^{ω}: q_{x,η} ≠ 0}. Thus w_{x} is finite and

x = ∑_{η∈w_{x}} q_{x,η}y_{η}.

The definition of the space T is considerably more technical since it reflects several different cases of the proof that the sets Ŝ_{x} are indeed finite. To this end let {q_{j}: j < ω} be a one-to-one enumeration of ℚ with q_{0} = 0. For i < ω let P_{i} = P({q_{j}: j < i}) and put P_{i} = P(2^{i} × {0, 1} × P_{i} × P_{i}). Note that each P_{i} is finite, so T = ∏_{i<ω} P_{i}, considered as the standard product of discrete spaces, is compact zerodimensional. We equip T with a distance function d defined between different s, t ∈ T by d(s, t) = 2^{− min{i<ω: s(i)≠t(i)}} and let

T_{0} = {t ∈ T: (∃n < ω)(∀i ≥ n) t(i) = ∅} .

Clearly T_{0} is countable.

Now we are ready to define g: ℝ → T_{0} ⊂ T. For this, however, we will need few more definitions. For x ∈ ℝ, q ∈ ℚ, i < ω, and ζ ∈ 2^{i} such that ζ ∈ {(η ↾ i): η ∈ w_{x}} we define:

• p(i) ∈ {0, 1} as the parity of i, i.e., p(i) = i mod 2;

• k_{i}(q) = {q_{j} ∈ ℚ: q_{j} < q & j < i} ∈ P_{i};

• η(x, ζ) to be the minimum of {η ∈ w_{x}: ζ ⊂ η} (in the lexicographical order);

• ξ(x, ζ) to be the minimum of {η ∈ w_{x}: ζ ⊂ η} \ {η(x, ζ)} provided |{η ∈ w_{x}: ζ ⊂ η}| ≠ 1; otherwise we put ξ(x, ζ) = η(x, ζ);

• n_{x} < ω to be the smallest number n > 0 such that

(i) η ↾ n ≠ ξ ↾ n for any different η, ξ ∈ w_{x}, and

(ii) q_{x,η} ∈ {q_{j}: j < n} for every η ∈ w_{x}.

Consider the function g: ℝ → T_{0} defined as follows. For every x ∈ ℝ and i < ω we define g(x)(i) ∈ P_{i} as

{⟨ζ, p(|{η ∈ w_{x}: ζ ⊂ η}|), k_{i}(q_{x,η(x,ζ)}), k_{i}(q_{x,ξ(x,ζ)})⟩: ζ ∈ {(η ↾ i): η ∈ w_{x}}}

provided i ≤ n_{x} and we put g(x)(i) = ∅ for n_{x} < i < ω. In the argument below the key role will be played by the function k_{i} in general, and the coordinate k_{i}(q_{x,η(x,ζ)}) in particular.

The key step in the proof that g has the desired property is that for every x ∈ ℝ and s ≠ 0

if n_{x} ≤ max{n_{x−s}, n_{x+s}} then g(x − s)(n_{x}) ≠ g(x + s)(n_{x}). (1)

To see (1) assume that n_{x} ≤ n_{x+s}. If n_{x−s} < n_{x} then g(x − s)(n_{x}) = ∅ ≠ g(x + s)(n_{x}), where g(x + s)(n_{x}) ≠ ∅ since w_{x+s} ≠ ∅ as n_{x−s} < n_{x} ≤ n_{x+s} implies x + s ≠ 0. Thus, we can assume that n_{x} ≤ min{n_{x−s}, n_{x+s}}. Take an η̂ ∈ w_{x−s} ∪ w_{x+s} such that q_{x−s,η̂} ≠ q_{x+s,η̂} and let ζ = η̂ ↾ n_{x}. Note that, by the definition of n_{x}, the set S = {η ∈ w_{x}: ζ ⊂ η} has at most one element.

If S = ∅ then {η ∈ w_{x−s}: ζ ⊂ η} = {η ∈ w_{x+s}: ζ ⊂ η} ≠ ∅ and so η(x−s, ζ) = η(x+s, ζ) ∉ w_{x} while q_{x−s,η(x−s,ζ)} +q_{x+s,η(x+s,ζ)} = 0. Thus q_{0} = 0 separates q_{x−s,η(x−s,ζ)} and q_{x+s,η(x+s,ζ)} implying that k_{n_{x}}(q_{x−s,η(x−s,ζ)}) ≠ k_{n_{x}}(q_{x+s,η(x+s,ζ)}). Therefore g(x − s)(n_{x}) ≠ g(x + s)(n_{x}).

So, assume that S ≠ ∅ and let η′ be the only element of S. Then η′ ∈ w_{x−s} ∪ w_{x+s}. If η′ belongs to precisely one of the sets w_{x+s} and w_{x−s}, say w_{x+s}, then {η ∈ w_{x+s}: ζ ⊂ η} = {η ∈ w_{x−s}: ζ ⊂ η} ∪ {η′}. In particular, p(|{η ∈ w_{x+s}: ζ ⊂ η}|) ≠ p(|{η ∈ w_{x−s}: ζ ⊂ η}|) implying that g(x−s)(n_{x}) ≠ g(x + s)(n_{x}).

So, we can assume that η′ ∈ w_{x−s} ∩ w_{x+s}. Then {η ∈ w_{x−s}: ζ ⊂ η} = {η ∈ w_{x+s}: ζ ⊂ η} and η(x − s, ζ) = η(x + s, ζ). We will consider three cases.

Case 1: η′ ≠ η(x − s, ζ) = η(x + s, ζ). Then q_{x−s,η(x−s,ζ)} + q_{x+s,η(x+s,ζ)} = 0, so q_{0} = 0 separates q_{x−s,η(x−s,ζ)} and q_{x+s,η(x+s,ζ)}. Thus k_{n_{x}}(q_{x−s,η(x−s,ζ)}) ≠ k_{n_{x}}(q_{x+s,η(x+s,ζ)}) and g(x − s)(n_{x}) ≠ g(x + s)(n_{x}).

Case 2: η′ = η(x − s, ζ) = η(x + s, ζ) and q_{x−s,η(x−s,ζ)} ≠ q_{x+s,η(x+s,ζ)}. Then q_{x−s,η(x−s,ζ)} + q_{x+s,η(x+s,ζ)} = 2q_{x,η′} and, by the definition of n_{x}, q_{x,η′} ∈ {q_{j}: j < n_{x}}. Since q_{x,η′} separates q_{x−s,η(x−s,ζ)} and q_{x+s,η(x+s,ζ)} we conclude that k_{n_{x}}(q_{x−s,η(x−s,ζ)}) ≠ k_{n_{x}}(q_{x+s,η(x+s,ζ)}) and g(x − s)(n_{x}) ≠ g(x + s)(n_{x}).

Case 3: η′ = η(x − s, ζ) = η(x + s, ζ) and q_{x−s,η(x−s,ζ)} = q_{x+s,η(x+s,ζ)}. Then Z = {η ∈ w_{x−s}: ζ ⊂ η}\{η(x−s, ζ)} = {η ∈ w_{x+s}: ζ ⊂ η}\{η(x+s, ζ)} is non-empty, since it contains η̂, and so ξ(x − s, ζ) = ξ(x + s, ζ) ∉ w_{x}. Therefore, as in Case 1, q_{x−s,ξ(x−s,ζ)} + q_{x+s,ξ(x+s,ζ)} = 0, so q_{0} = 0 separates q_{x−s,ξ(x−s,ζ)} and q_{x+s,ξ(x+s,ζ)}. Thus k_{n_{x}}(q_{x−s,ξ(x−s,ζ)}) ≠ k_{n_{x}}(q_{x+s,ξ(x+s,ζ)}) and g(x − s)(n_{x}) ≠ g(x + s)(n_{x}).

This finishes the proof of (1).

Next, for every x ∈ ℝ put δ_{x} = 2^{−n_{x}}. To finish the proof of the theorem it is enough to show that every Ŝ_{x} defined for such a choice of δ_{x} is a subset of a finite set

Z_{x} = {s ∈ ℝ: w_{x+s} ⊂ w_{x} & n_{x+s} < n_{x}} = { ∑_{η∈w_{x}} p_{η}y_{η}: p_{η} ∈ {q_{j}: j < n_{x}}} .

Indeed, take an s ∈ Ŝ_{x}. Then, by (1) and the definition of the distance function d, we have max{n_{x−s}, n_{x+s}} < n_{x}. Notice also that if n_{x−s} ≠ n_{x+s}, say n_{x−s} < n_{x+s}, then g(x − s)(n_{x+s}) = ∅ ≠ g(x + s)(n_{x+s}) implying that d(g(x + s), g(x − s)) ≥ 2^{−n_{x+s}} > 2^{−n_{x}} = δ_{x}, which contradicts s ∈ Ŝ_{x}. So, we have n_{x−s} = n_{x+s}. To prove that s ∈ Z_{x} it is enough to show that w_{x+s} ⊂ w_{x}. But if it is not the case then there exists an η ∈ w_{x+s} \ w_{x}. Moreover, q_{x+s,η} = −q_{x−s,η} ≠ 0 and η = η(x + s, ζ) = η(x − s, ζ), where ζ = η ↾ n_{x+s}. In particular, q_{0} = 0 separates q_{x+s,η(x+s,ζ)} and q_{x−s,η(x−s,ζ)}. Therefore k_{n_{x+s}}(q_{x−s,η(x−s,ζ)}) ≠ k_{n_{x+s}}(q_{x+s,η(x+s,ζ)}) and g(x − s)(n_{x+s}) ≠ g(x + s)(n_{x+s}). So d(g(x + s), g(x − s)) ≥ 2^{−n_{x+s}} > 2^{−n_{x}} = δ_{x} again contradicting s ∈ Ŝ_{x}. Thus, w_{x+s} ⊂ w_{x} and s ∈ Z_{x}. □

References

[1] K. Ciesielski, On range of uniformly antisymmetric functions, Real Analysis Exch. 19(2) (1993–94), 616–619.

[2] K. Ciesielski, Uniformly antisymmetric functions and K_{5}, Real Analysis Exch. 21(2) (1995–96), 147–153. (Preprint^{⋆} available.^{1})

[3] K. Ciesielski, Sum and difference free partitions of vector spaces, Colloq. Math. 71 (1996), 263-271. (Preprint^{⋆} available.)

[4] K. Ciesielski, Set Theory for the Working Mathematician, London Math. Soc. Student Texts 39, Cambridge Univ. Press 1997.

[5] K. Ciesielski, Set theoretic real analysis, J. Appl. Anal. 3(2) (1997), 143– 190. (Preprint^{⋆} available.)

[6] K. Ciesielski, L. Larson, Uniformly antisymmetric functions, Real Analysis Exch. 19(1) (1993–94), 226–235.

[7] P. Kostyrko, There is no strongly locally antisymmetric set, Real Analysis Exch. 17 (1991–92), 423–425.

[8] P. Komjáth, S. Shelah, On uniformly antisymmetric functions, Real Analysis Exch. 19(1) (1993–94), 218–225.

[9] Brian Thomson, Symmetric Properties of Real Functions, Marcel Dekker, 1994.

1Preprints marked by ^{⋆} are available in electronic form from Set Theoretic Analysis Web Page: http://www.math.wvu.edu/homepages/kcies/STA/STA.html