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PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 75, Number 2, July 1979
SOME RESULTS CONNECTED WITH
A PROBLEM OF ERDÖS. Ü
HARRY I. MILLER1
Abstract. It is shown, using the continuum hypothesis, that if E is an
uncountable subset of the real line, then there exist subsets 5, and S2 of the
unit interval, such that S¡ has outer Lebesgue measure one and 52 is of the
second Baire category and such that neither Sx nor S2 contains a subset
similar (in the sense of elementary geometry) to E. These results are related
to a conjecture of P. Erdôs.
1. Introduction. P. Erdôs [2] presented the following conjecture at the
problem session of the Fifth Balkan Mathematical Congress (Belgrade, June
24-30, 1974):
Conjecture. Let E be an infinite set of real numbers. Then there exists a set
of real numbers S of positive Lebesgue measure which does not contain a set
£" similar (in the sense of elementary geometry) to E.
If E is a finite set of real numbers, then every set of real numbers S of
positive Lebesgue measure contains a subset E' similar to E. This follows
from a result of M. S. Ruziewicz [5] or as P. Xenikakis has shown (in a private
communication) from Theorem 3 in [3]. By Theorem 4 in [3], the corre-
sponding result holds for Baire sets S of the second Baire category (S is a
Baire set if it can be written in the form S = (G \ C) u D, where G is an
open set and C and D are sets of the first Baire category).
H. I. Miller and P. Xenikakis [4] have proven the following two theorems
related to the conjecture of Erdôs.
Theorem A. If A ci (the real line) possesses the Baire property and is of
the second Baire category in R and if (£„),?_ x is a convergent sequence of reals,
then A contains a set A' which is similar to the set [zn; n = 1, 2, . . . }.
Theorem B. If A c R can be written in the form A = (G \ C) U D, where
G is a nonempty open set and C and D are sets of Lebesgue measure zero and if
(z„)n°=x is a convergent sequence of reals, then A contains a set A' which is
similar to the set {z„; n = 1, 2, . . . }.
The purpose of this work is to show, using transfinite induction and the
continuum hypothesis, that if E is an uncountable subset of the real line, then
Received by the editors May 1, 1978.
AMS (A/OS) subject classifications (1970). Primary 28A05, 26A21.
'The work on this paper was supported by the Scientific Research Fund of Bosna and
Herzegovina.
© 1979 American Mathematical Society
0002-9939/79/0000-031 S/S02.00
265
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266 H. I. MILLER
there exist subsets S, and 52 of the interval [0, 1] such that m*(Sx) = 1 and S2
is of the second Baire category in R, and such that neither 5, nor S2 contains
a subset similar to E. Here m* denotes outer Lebesgue measure.
2. Results. Our first theorem makes use of the following lemma whose
proof can be found in [1].
Lemma A. Let C be a closed subset of the real line. Let B be a subset of C
such that B has nonempty intersection with every closed subset of C of positive
Lebesgue measure. Then m*(B) = m(C).
We now proceed to prove our first theorem.
Theorem 1. If E is an uncountable set of real numbers, then there exists a
subset S of [0, 1] such that m*(S) = 1 and no subset of S is similar to E.
Proof. Let 2 - [Q; Q a closed subset of [0, 1] and m(Q) > 0}. By the
continuum hypothesis 2 can be written in the form 2 = {o„;a<fi} where
ß denotes the first uncountable ordinal. Similarly S, the family of all subsets
of R similar to E, can be written in the form $ = [Ea; a < ß}. This is true
as there are c, the cardinality of the continuum, similarity transformations;
since each similarity transformation / is of the form/(x) = ax + b. We now
proceed to construct two transfinite sequences {*a}a<a and {^a}a<ß of real
numbers.
Pick je, G Qx andyx G Ex such that xx ¥=yx- Suppose that w is an ordinal
number u < ß and that {*a}a<w and {ya)a<a have been selected such that:
(a) xa E Qa,ya E Ea for each a < w, and
(b) [xa; a< ß)n{ya;a< ß) =0, for every ß, ß < co.
Then we can find xu E Qa and yu G Ea such that {xa; a < w} n [ya;
a < w} is the empty set, since Qu and Ea are uncountable sets and u < ß
implies that w is a countable ordinal. Therefore by transfinite sequences
{xa}a<a aQd 0„L<a such to-t xa E Qa, ya E Ea for each a,a<Ü, and
such that {xa; a < ß) n [ya; a < ß) =0 for every ß, ß < ß. Let S denote
the set [xa; a < Q). Then S c [0, 1] and by Lemma A we have m*(S) = 1.
Furthermore, no subset of S is similar to E. For if some subset, say S',of S is
similar to E we have S' = Ey for some y < fi. This in turn implies yy G S'
and hence yy G S. Therefore there exists 5 < ß such that yy = xs or [xa;
a < ß) n [ya; a < ß] ¥=0, where ß = max(y, 5), which is a contradiction.
We need the following lemma in the proof of Theorem 2.
Lemma B. Let 2 denote the collection of subsets of[0, 1] given by the formula
2 = {(/? \ U" i F¡) n [0, 1] where each F¡ is a closed and nowhere dense
subset of R). If B is a subset of [0, 1] and has the property that Q n B =£0 for
every g G 2, then B is a set of the second Baire category in R.
Proof. If B is a set of the first Baire category, then we have B = U"i X¡,
with each X¡ nowhere dense in R. This implies that fie U"i Cl( X¡), where
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SOME RESULTS CONNECTED WITH A PROBLEM OF ERDÖS. II 267
Cl denotes the closure operator. From this it follows that fln(Ä\
U," i C1(A",)) =0, contradicting the assumption that Q n B ¥=0 for every
0<ES.
Theorem 2. If E is an uncountable set of real numbers, then there exists a
subset S of [0, 1] with the property that no subset of S is similar to E and such
that S is of the second Baire category in R.
Proof. The proof of Theorem 2 is essentially the same as that of Theorem
1. Here we make use of Lemma B and the fact that the family of sets 2. given
in Lemma B can, by assuming the continuum hypothesis, be written in the
form 2 = {g,;a<Q}, where as before, ß denotes the first uncountable
ordinal.
3. Remark. Professor John C. Oxtoby has observed that Theorems 1 and 2
can be obtained by applying a theorem of Sierpiñski [6] which reads:
Theorem C. Given a set X, a family $ of subsets of X, and a group G of 1-1
mappings of X onto itself, such that card X = card Í» = card G = 3C, and such
that X \ U¡fi(H¡) is uncountable for each sequence {/,} c G and {//,} c $,
then there exists an uncountable set S C X with the properties:
H E $ implies H n S is countable, and
f E G implies f(S) \ S is countable.
We will sketch Oxtoby's proof. In the following, let £ = {L; L an uncount-
able subset of R that contains no uncountable meager subset} and S = (S;
S an uncountable subset of R that contains no uncountable subset of measure
zero}. Furthermore let X = [0, 1], ¥, denote all Gs nullsets contained in X,
^2 denote all F0 meager subsets of X, Gx denote all 1-1 Borel measurable and
nullset-preserving transformations of X, and G2 denote all 1-1 Borel
measurable and category-preserving transformations of X. Let S denote all
sets similar to a given fixed uncountable subset of R. Define
$, = *, U [E E & ; E c X and E E S }
and
$2 = *2 u [E E & ; E c X and E E £}.
It is easy to verify, by assuming the continuum hypothesis, that the hypothe-
ses of Sierpiñski's theorem are satisfied when we take G = Gx and $ = $,.
The resulting set Sx c [0, 1] has the following properties:
(i) Sx G S and hence is nonmeasurable and of the first category on every
perfect set;
(ii) Sx contains no member of & ;
(iii)/ G Gx implies f(Sx)ASx is countable, and
(iv) m*(Sx) = 1.
Similarly, taking G = G2 and $ = i»2 and by assuming the continuum
hypothesis we obtain a set S2 c [0, 1] with the following properties:
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268 H. I. MILLER
(i) S2 G £ and hence is of measure zero and does not possess the Baire
property;
(ii) S2 contains no member of S ;
(iii)/ G G2 implies f(S2)AS2 is countable, and
(iv) S2 is of the second category at each point of [0, 1].
References
1. A. Abian, Partition of nondenumerable closed sets of reals, Czechoslovak Math. J. 26(101)
(1976), no. 2, 207-210. MR 53 #5820.
2. P. Erdôs, Problems, Math. Balkánica (Papers presented at The Fifth Balkan Mathematical
Congress), 4 (1974), 203-204.
3. H. I. Miller, Relationships between various gauges of the size of sets of real numbers. II, Akad.
Nauka i Umjet. Bosne i Hercegov. Rad. Odjelj. Prirod. Mat. Nauka 59 (1976), 37-48.
4. H. I. Miller and P. I. Xenikakis, Some results connected with a problem of Erdôs. I, Akad.
Nauka i Umjet. Bosne i Hercegov. Rad. Odjelj. Prirod.' Mat. Nauka (to appear).
5. M. S. Ruziewicz, Contribution à l'étude des ensembles de distances de points, Fund. Math. 7
(1925), 141-143.
6. W. Sierpiñski, Un théorème de la théorie générale des ensembles et ses applications, C. R. Soc.
Sei. Varsovie 28 (1936), 131-135. Zbl. 15, 103.
Department of Mathematics, University of Sarajevo, Sarajevo 71000, Yugoslavia
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