Covering the real line with translates of a zero dimensional compact set

András Máthé^{∗}

Abstract

We construct a compact set C of Hausdorff dimension zero so that cof(N) many translates of C cover the real line. Hence it is consistent with ZFC that less than continuum many translates of a zero dimensional compact set can cover the real line. This answers a question of Dan Mauldin.

1 Introduction

Given a set A ⊂ ℝ, how many of its translates are needed to cover the real line? It is clear that if A is of Lebesgue measure zero or if A is of the first Baire category, then countable many translates are not enough. Therefore if we assume the continuum hypothesis, then for these kind of sets clearly we need continuum many translates to cover the real line.

Gary Gruenhage showed that it is not possible to cover ℝ with less than continuum many translates of the standard middle-third Cantor set. (This is a proof in ZFC, without any extra set theoretic assumptions.) He then raised the following question (see also [2] and [3]):

Is it consistent that there exist a compact set of Lebesgue measure zero and less than continuum many of its translates that cover ℝ?

Since this was unsolved for some time, Dan Mauldin asked whether at least the compact sets of Hausdorff dimension less than 1 possess the property that less than continuum many of their translates are not enough to cover ℝ. Udayan Darji and Tamás Keleti [2] showed in a beautiful simple proof that this is indeed true for compact sets of packing dimension less than 1.

Márton Elekes and Juris Steprāns solved the question of Gruenhage [3]. They showed that ℝ can be covered by cof(N) many translates of some compact set of measure zero of Erdős and Kakutani. Here cof(N) is the so-called cofinality invariant of Lebesgue null sets, and it is consistent that cof(N) is less than continuum (see e.g. [1]).

In this note we answer Dan Mauldin’s question in the negative. We present a compact set C of Hausdorff dimension zero so that ℝ can be covered by cof(N) many translates of C (Theorem 2.2). Thus it is consistent with ZFC that ℝ can be covered by less than continuum many translates of some compact set of Hausdorff dimension zero.

We remark here that if we do not restrict ourselves to compact sets then the previous statement is clear: If A is a residual subset of ℝ, and B ⊂ ℝ is a set of the second Baire category, then clearly A + B = ℝ. Let us choose A to be a residual G_{δ} set of Hausdorff dimension zero (e.g. the set of Liouville numbers). It is consistent that there exists a set of the second Baire category consisting of less than continuum many points (see e.g. [1]), that set should be chosen as B. Therefore it is consistent that ℝ can be covered by less than continuum many translates of a zero dimensional G_{δ} set.

In the next section we review how it is possible to write ℝ as a union of cof(N) many “small” compact sets and we state our main results. These are then proved in the third section.

∗Supported in part by the Hungarian Scientific Research Fund grant no. T 72655.

MSC Codes: Primary 28A78; Secondary 03E17, 03E35 Key words: Hausdorff dimension, consistent, continuum, cover, translate

Notation and definitions. We denote the cofinality invariant of Lebesgue null sets by cof(N), which is the smallest cardinal κ for which there exist κ many null sets so that every null set is contained in one of them.

We denote the d-dimensional Hausdorff pre-measure of a set A ⊂ ℝ by H_{∞}^{d}(A), defined as

H_{∞}^{d}(A) = inf{∑_{i=1}^{∞}(diam I_{i})^{d} : A ⊂ ∪_{i=1}^{∞}I_{i}}.

The Hausdorff dimension of A is then defined as inf{d > 0 : H_{∞}^{d}(A) = 0}.

We denote the set of real numbers by ℝ and the set of integers by ℤ.

2 Covering ℝ with small sets

Let ω denote the first infinite ordinal, that is, ω = {0, 1, 2, …}. Let f : ω \ {0} → ω \ {0, 1}. A set of the form S = ∏_{n=1}^{∞} A_{n}, where each A_{n} is of size at most f(n), is called an f-slalom (by ∏ we mean the direct product of sets). It is known that if lim f(n) = ∞, then ω^{ω} can be covered by cof(N) many f-slaloms (Gruenhage and Levy [4, Theorem 2.12.] and [5], see also Bartoszyński and Judah [1]).

Hence it is clear that if we fix two functions f, g : ω\{0} → ω\{0, 1} where lim g(n) = ∞, then every f-slalom can be covered by cof(N) many g-slaloms. We describe below how it is possible to use this fact to cover the real line with cof(N) many “small” compact sets (here we follow Elekes and Steprāns [3]). So let us take the f-slalom T = ∏_{n=1}^{∞}{0, 1, …, f(n)−1}, and let us cover it by cof(N) many g-slaloms S_{α} (α < cof(N)). We may suppose without loss of generality that each S_{α} is contained in T; that is, they are of the form ∏_{n=1}^{∞} A_{n} where A_{n} ⊂ {0, 1, …, f(n) − 1} and |A_{n}| ≤ g(n) for each n. Then

[0, 1] = \{∑_{n=1}^{∞} \frac{s_{n}}{f(1)f(2) … f(n)} : (s_{n})_{n=1}^{∞} ∈ T \}.

(It is worth thinking of s_{n} as digits of a numeral system with “increasing base” corresponding to the series f(n).) For a slalom S ⊂ T define

S^{∗} = \{∑_{n=1}^{∞} \frac{s_{n}}{f(1)f(2) … f(n)} : (s_{n})_{n=1}^{∞} ∈ S\},  (1)

or equivalently, when S = ∏_{n=1}^{∞} A_{n},

S^{∗} = (∏_{n=1}^{∞} A_{n})^{∗} = \frac{1}{f(1)}A_{1} + \frac{1}{f(1)f(2)}A_{2} + \frac{1}{f(1)f(2)f(3)}A_{3} + ….  (2)

(This infinite Minkowski sum is well-defined, and as S^{∗} is a continuous image of the compact space ∏ A_{n}, it is compact. For later reference note that (∏_{n=1}^{∞} A_{n})^{∗} makes sense even if, say, A_{n} ⊂ {0, 1, …, 2f(n) − 1}.)

Since the slaloms S_{α} cover T, the compact sets S_{α}^{∗} (α < cof(N)) cover the unit interval. These sets S_{α}^{∗} are small if g is much smaller than f. That is, as it is easy to see, the Hausdorff dimension of the sets S_{α}^{∗} is at most

lim inf_{n→∞} \frac{log(g(1) … g(n))}{log(f(1) … f(n))},

which can be zero if we choose g and f accordingly. Thus [0, 1] (and also of course ℝ) can be covered by cof(N) many compact sets of Hausdorff dimension zero. However, our point here is to find one compact set of Hausdorff dimension zero so that ℝ can be covered by cof(N) many translates of this compact set. What we will do is to find a compact set C of Hausdorff dimension zero so that for every g-slalom S ⊂ T, a translate of C will cover S^{∗}. Then clearly ℝ can be covered by cof(N) many translates of C.

We will choose g to be (say) g(n) = n + 1, the function f will be determined implicitly by the construction.

Theorem 2.1. Let g(n) = n+1. There exists a compact set C of Hausdorff dimension zero and a function f : ω \{0} → ω \{0, 1} so that for every g-slalom S ⊂ ∏_{n=1}^{∞}{0, 1, …, f(n)− 1}, S^{∗} can be covered by a translate of C.

We prove this theorem in the next section. As it was shown above, this theorem immediately yields our main result.

Theorem 2.2. There exists a compact set C of Hausdorff dimension zero so that ℝ can be covered by cof(N) many translates of C.

3 Construction

Before we prove Theorem 2.1, we outline the construction of the set C in an informal way. So we need to construct a (small) set C so that every (very small) set S^{∗} ⊂ [0, 1] can be covered by a translate of C. First we will find compact sets K_{n} ⊂ [0, 2] (each of which is a union of finitely many intervals) with arbitrarily small Hausdorff pre-measure (of arbitrarily small dimension) so that every (n + 1)-point set B ⊂ [0, 1] can be covered by a translate of K_{n}. Then to this set K_{n} we associate a large positive integer f(n) and a set A_{n} ⊂ {0, 1, …, 2f(n) − 1} with the property that every set B_{n} ⊂ {0, 1, …, f(n) − 1} with |B_{n}| ≤ n + 1 can be covered by a translate of A_{n}. Setting g(n) = n + 1, this yields that for every g-slalom ∏ B_{n} the set (∏ B_{n})^{∗} (as defined by (2)) can be covered by a translate of (∏ A_{n})^{∗}. Finally, we show that (∏ A_{n})^{∗} is of dimension zero using the fact that the sets K_{n} can be chosen to be “arbitrarily small”. Therefore we can let C to be (∏ A_{n})^{∗}.

Lemma 3.1. For every d, δ > 0 and for every positive integer n there exist a compact set K_{n} ⊂ [0, 2] and some ε_{n} > 0 so that H_{∞}^{d}(K_{n}) < δ holds and that for every set B ⊂ [0, 1] with |B| = n + 1 there exists some t ∈ ℝ for which B + [t, t + ε_{n}] ⊂ K_{n} holds.

Proof. We prove this lemma by induction on n. We start with the case n = 1.

Let a, b > 0, let

K_{1} = [0, a] ∪ ([0, 2] ∩ ∪_{i∈ℤ}[ia, ia + b]).

It is easy to check that if B ⊂ [0, 1], |B| = 2, then there exists some t ∈ ℝ so that B + t ⊂ K_{1}. Moreover, we have some freedom in choosing t; that is, there exists some ε_{1} > 0 independent of B such that B + [t, t + ε_{1}] ⊂ K_{1} for some t (in fact, ε_{1} can be chosen to be b). The set K_{1} can be covered by an interval of length a and at most 2/a + 1 intervals of length b, thus H_{∞}^{d}(K_{1}) ≤ a^{d} + (2/a + 1)b^{d}. Therefore if we choose a small enough, and then b small enough, H_{∞}^{d}(K_{1}) can be arbitrarily small.

Now let us suppose that the lemma holds for some n, we prove it for n + 1. So if we want a set K_{n+1} with H_{∞}^{d}(K_{n}) < δ for some d and δ, let us start with choosing a set K_{n} with H_{∞}^{d}(K_{n}) < δ/2 and an ε_{n} > 0 so that for every set B ⊂ [0, 1] with |B| = n + 1 we have B + [t, t + ε_{n}] ⊂ K_{n} for some t.

Now let 0 < b < ε_{n}/2 be so small that (2/ε_{n} + 1)(2b)^{d} < δ/2. Let

K_{n+1} = K_{n} ∪ ([0, 2] ∩ ∪_{i∈ℤ}[iε_{n} − b, iε_{n} + b]).

Then we have H_{∞}^{d}(K_{n+1}) < δ.

Let B ⊂ [0, 1] and |B| = n + 2. We show that B can be translated into the set K_{n+1}. First let x be some element of B which is not the smallest or the largest one. Let B′ = B \ {x}. Then there exists some t′ so that B′ + [t′, t′ + ε_{n}] ⊂ K_{n}. It is easy to check that there exists some t so that [t, t + b] ⊂ [t′, t′ + ε_{n}] and x + [t, t + b] ⊂ K_{n+1}. Then clearly B + [t, t + b] ⊂ K_{n+1}. Thus ε_{n+1} can be chosen to be b.

Let us fix some K_{1} and ε_{1} (so that the statement of Lemma 3.1 holds) with H_{∞}^{1}(K_{1}) < 1. Then let us fix some K_{2} and ε_{2} so that H_{∞}^{1/2}(K_{2}) < 100^{−1}ε_{1}. If K_{1}, …, K_{n} and ε_{1}, …, ε_{n} are already fixed, then let us fix K_{n+1} and ε_{n+1} so that the statement of Lemma 3.1 holds with

H_{∞}^{1/(n+1)}(K_{n+1}) < 100^{−n}ε_{1}ε_{2} … ε_{n}.  (3)

The choice of these bounds will be clear only later.

To each of these sets K_{n} and ε_{n} we associate a set A_{n} ⊂ ℤ and some f(n) ∈ ω in the following way. Let f(n) be a positive integer so that

\frac{5}{ε_{n}} < f(n) < \frac{10}{ε_{n}}  (4)

holds. Let

A_{n} = \{i ∈ ℤ : [\frac{i}{f(n)}, \frac{i+4}{f(n)}] ⊂ K_{n}\} ⊂ \{0, 1, …, 2f(n) − 1\}.  (5)

(Note that this definition implies \frac{1}{f(n)}A_{n} + [0, \frac{4}{f(n)}] ⊂ K_{n}, which we will use (only) in Claim 3.4.)

Claim 3.2. If B_{n} ⊂ {0, 1, …, f(n) − 1} and |B_{n}| = n + 1, then there exists some u ∈ ℤ so that B_{n} + u ⊂ A_{n}.

Proof. By Lemma 3.1 there exists some t so that B_{n}/f(n) + [t, t + ε_{n}] ⊂ K_{n}, thus B_{n} + [tf(n), tf(n)+ε_{n}f(n)] ⊂ f(n)K_{n}. Using (4), the length of the interval [tf(n), tf(n)+ε_{n}f(n)] is at least 5, so it has a subinterval of the form [u, u+4] with integer u. Thus B_{n}+[u, u+4] ⊂ f(n)K_{n}. Then B_{n} + u ⊂ A_{n}.

Let

C = (∏_{n=1}^{∞} A_{n})^{∗} = \frac{1}{f(1)}A_{1} + \frac{1}{f(1)f(2)}A_{2} + \frac{1}{f(1)f(2)f(3)}A_{3} + …,  (6)

thus C is compact, see our remark after definition (2).

Claim 3.3. Let g(n) = n + 1. For every g-slalom S ⊂ ∏_{n=1}^{∞}{0, 1, …, f(n) − 1}, S^{∗} can be covered by a translate of C.

Proof. Let B_{n} ⊂ {0, 1, …, f(n) − 1}, |B_{n}| ≤ n + 1 be arbitrary (n = 1, 2, …). Let S be the g-slalom ∏_{n≥1} B_{n}. Then by definition (2), we have

S^{∗} = \frac{1}{f(1)}B_{1} + \frac{1}{f(1)f(2)}B_{2} + \frac{1}{f(1)f(2)f(3)}B_{3} + ….

We can choose an integer u_{n} such that B_{n} + u_{n} ⊂ A_{n} by Claim 3.2. Then from (6),

S^{∗} + \frac{u_{1}}{f(1)} + \frac{u_{2}}{f(1)f(2)} + … ⊂ C.

In order to finish the proof of Theorem 2.1, what is left is to show the following.

Claim 3.4. The Hausdorff dimension of C is zero.

Proof. For n ≥ 1 let

C_{n} = \frac{1}{f(1)f(2) … f(n)}A_{n} + \frac{1}{f(1)f(2) … f(n + 1)}A_{n+1} + ….  (7)

Thus C = C_{1} and C is the union of |A_{1}||A_{2}| … |A_{n−1}| many translated copies of C_{n}. First we show that C_{n} lies in a small neighbourhood of the first term on the right hand side of (7). For k ≥ 1 we have

sup \frac{1}{f(n)f(n + 1) … f(n + k)}A_{n+k} ≤ \frac{2}{f(n)f(n + 1) … f(n + k − 1)} ≤ \frac{2}{f(n)2^{k−1}} ,

thus

sup f(1)f(2) … f(n − 1)C_{n+1} ≤ ∑_{k=1}^{∞} \frac{2}{f(n)2^{k−1}} = \frac{4}{f(n)},

and

f(1)f(2) … f(n − 1)C_{n+1} ⊂ [0, \frac{4}{f(n)}].  (8)

Since (7) implies C_{n} = \frac{1}{f(1)f(2)...f(n)}A_{n} + C_{n+1}, using (8) we obtain

f(1)f(2) … f(n − 1)C_{n} ⊂ \frac{1}{f(n)}A_{n} + [0, \frac{4}{f(n)}].

From the definition of A_{n} (5) we see that the right hand side is contained in K_{n}, thus

f(1)f(2) … f(n − 1)C_{n} ⊂ K_{n}.

Hence

H_{∞}^{d}(C_{n}) ≤ (f(1)f(2) … f(n − 1))^{−d}H_{∞}^{d}(K_{n}) ≤ H_{∞}^{d}(K_{n}).

As we have already mentioned, C is the union of |A_{1}||A_{2}| … |A_{n−1}| many translated copies of C_{n}, therefore

H_{∞}^{1/n}(C) ≤ |A_{1}||A_{2}| … |A_{n−1}|H_{∞}^{1/n}(C_{n})

≤ |A_{1}||A_{2}| … |A_{n−1}|H_{∞}^{1/n}(K_{n}) ≤ (2f(1))(2f(2)) … (2f(n − 1))H_{∞}^{1/n}(K_{n}).

Using (3) and the upper bound for f(n) in (4) we obtain

H_{∞}^{1/n}(C) ≤ 20^{n−1} \frac{1}{ε_{1} … ε_{n−1}} 100^{−(n−1)}ε_{1} … ε_{n−1} = 5^{−(n−1)}.

Thus for all d > 0 for all n > 1/d we have H_{∞}^{d}(C) ≤ H_{∞}^{1/n}(C) ≤ 5^{−(n−1)}, therefore H_{∞}^{d}(C) = 0. So the Hausdorff dimension of the set C is zero.

Acknowledgment. The author is grateful to Márton Elekes for his valuable remarks.

References

[1] T. Bartoszyński, H. Judah, Set theory: On the structure of the real line, A. K. Peters, Wellesley, Massachusetts, 1995.

[2] U. Darji, T. Keleti, Covering ℝ with translates of a compact set, Proc. Amer. Math. Soc. 131 (2003), 2593–2596.

[3] M. Elekes, J. Steprāns, Less than continuum many translates of a compact nullset may cover the real line, Fund. Math. 181 (2004), no. 1, 89–96.

[4] G. Gruenhage, R. Levy, Covering ^{ω}ω by special Cantor sets, Comment. Math. Univ. Carolin. 43 (2002), no. 3, 497–509.

[5] G. Gruenhage, R. Levy, Correction to the paper: “Covering ^{ω}ω by special Cantor sets”, Comment. Math. Univ. Carolin. 45 (2004), no. 1, 177.

Alfréd Rényi Institute of Mathematics Reáltanoda u. 13-15 1053 Budapest, Hungary

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