REPORTS ON REAL ANALYSIS
XXIII International Conference
on Real Functions Theory
Niedzica 2009,
pp. XXX–xxx
Thin sets of reals related to trigonometric series
Peter Eliaš1
1. Introduction
In 1812, J. Fourier came with an opinion that every function f : [0, 1] → R
can be expressed as a sum of trigonometric series
∞
∑
(an cos nx + bn sin nx).
(1)
n=0
Fortunately (for the later extensive development of trigonometric series theory), he was wrong. Nevertheless, something very close to Fourier’s idea is
still valid: for every nice function f there exists a trigonometric series (1)
such that the equality
f (x) =
∞
∑
(an cos nx + bn sin nx)
(2)
n=0
holds true for every x, with some small set of exceptions. As an example we
can mention the following statement, conjectured by N. N. Luzin in 1915
and proved by L. Carleson in 1966.
Theorem 1.1. If a 2π-periodic function f is square-integrable, i.e., it belongs to the space L2 , then its Fourier series is such that the set of all points
x ∈ R for which the equality (2) fails has Lebesgue measure zero.
In 1912, A. Denjoy and N. N. Luzin independently proved the following
theorem.
1
Author was supported by grant VEGA 1/0032/09 of Slovak Grant Agency.
2
Peter Eliaš
Theorem 1.2. If a trigonometric series (1) absolutely converges on a set
∞
∑
of positive Lebesgue measure then
(|an | + |bn |) < ∞, and hence the series
n=0
(1) converges everywhere.
N. N. Luzin also proved an analogous theorem for sets having Baire property which are of second category (i.e., non-meager). The converse to these
two theorems is not valid: there exists a set X which is both of first category and of zero Lebesgue measure such that every trigonometric series
absolutely converging on X is absolutely converging everywhere, e.g., the
standard Cantor set.
In 1938, J. Marcinkiewicz introduced the following notion.
Definition 1.3. A set X ⊆ R is a set of absolute convergence if there exists
a trigonometric series (1) absolutely converging on X which is not absolutely
converging everywhere.
Sets of absolute convergence are also called N-sets, in the honour of V.
V. Niemytzki. The theorems of Denjoy and Luzin say that N-sets are of
Lebesgue measure zero and of first category. It is also not difficult to show
that every N-set is included in an Fσ N-set which is a subgroup of R, every
countable set is N-set, and that there exist perfect N-sets. Moreover, a
linear transformation of an N-set is still an N-set. The family of all N-sets
is hereditary (a subset of an N-set is an N-set) but is not an ideal: there
exist two perfect N-sets X, Y such that the group generated by X ∪ Y is
the whole real line, and hence X ∪ Y cannot be an N-set.
N-sets were thoroughly investigated in works of R. Salem [12] and J. Arbault [1].
2. Families of trigonometric thin sets
Since trigonometric functions are periodic, it is convenient to work on
the unit circle T instead of the real line R. We prefer the additive notation,
i.e., T is the quotient topological group R/Z with respect to the operation
+. We do not distinguish between the elements of T and points on the real
line.
Let ∥x∥ denote the distance of x ∈ T to 0, or, when working on the real
line, a distance of x ∈ R to the nearest integer.
The following characterization of N-sets was proved by R. Salem [12].
Theorem 2.1. A set X ⊆ T is an N-set iff there exists a sequence {an }∞
n=1
∞
∞
∑
∑
of non-negative reals such that
an = ∞ and for x ∈ X,
an ∥nx∥ < ∞.
n=1
n=1
The proof is not difficult and is based on the use of Dirichlet principle.
Thin sets of reals related to trigonometric series
3
Many other kinds of sets related to the convergence of trigonometric sets
were defined and investigated (see, e.g, [3]). We mention just few of them.
Definition 2.2. A set X ⊆ T is called
• an N0 -set if there exists an increasing sequence {nk }∞
k=0 such that
∞
∑
∥nk x∥ < ∞,
k=0
• a Dirichlet set (also D-set) if there exists an increasing sequence
{nk }∞
k=0 such that ∥nk x∥ ⇒ 0 on X,
• a pseudo-Dirichlet set (also pD-set) if there exists an increasing se−k
quence {nk }∞
k=0 such that (∀x ∈ X) (∃K) (∀k ≥ K) ∥nk x∥ ≤ 2 ,
• an Arbault set (also A-set) if there exists an increasing sequence
{nk }∞
k=0 such that lim ∥nk ∥ = 0 for all x ∈ X.
k→∞
We denote by N , N0 , D, pD, A the families of all N-sets, N0 -sets, D-sets,
pD-sets, and A-sets, respectively.
It can be seen directly from the definition that D ⊆ pD ⊆ N0 ⊆ N and
N0 ⊆ A, and all families, except D, are generated by proper Borel subgroups
of T. It is well known that no other inclusion between these families holds
true (see e.g. [3]). Moreover, the examples of sets distinguishing these
families can be expressed in a very general form [4].
Theorem 2.3. Let {nk }∞
k=0 be an increasing sequence of natural numbers,
and let {ak }∞
be
a
sequence
of non-negative reals.
k=0
{
}
∞
∞ n
∑
∑
k
< ∞ then x ∈ T :
1. If
∥nk x∥ < ∞ ∈ N0 \ pD.
k=0
k=0 nk+1
{
}
nk
2. If lim
= 0 then x ∈ T : lim ∥nk x∥ = 0 ∈ A \ N .
k→∞ nk+1
k→∞
∞
∞
∑
∑
nk
< ∞,
3. If lim ak = 0,
ak = ∞, and
ak
k→∞
nk+1
k=0
k=0
}
{
∞
∑
ak ∥nk x∥ < ∞ ∈ N \ A.
then x ∈ T :
k=0
However, not much is known about sets from the above theorem in the
case when the corresponding conditions are not fulfilled.
3. Inclusions between Arbault sets
In this section we consider two Arbault sets defined by different sequences
of natural numbers and try to answer the question whether they are in an
inclusion or not.
For a given sequence a = {an }n∈N , let us denote
{
}
A(a) = x ∈ T : lim an ∥nx∥ = 0 .
n→∞
4
Peter Eliaš
The following problem is open for a long time. As I have learned from
prof. V. Sós, this problem was treated by D. Maharam and A. Stone.
Problem 3.1. Characterize those sequences a for which the set A(a) is
countable.
In Theorem 3.3 we give a partial answer to a related question: when does
the inclusion A(a) ⊆ A(b) hold true?
The following notion was implicitly introduced in [5].
Definition 3.2. Let k ∈ N, and let z = {zm,n }m,n∈N be an infinite matrix
of integers. We say that z is a k-bounded matrix if
1. (∀n) (∃M ) (∀m > M ) zm,n = 0, and
∞
∑
2. (∀m)
|zm,n | ≤ k,
n=0
z is a bounded matrix if it is a k-bounded matrix for some k.
Theorem 3.3. Let a = {a}n∈N and b = {bn }n∈N be increasing sequences of
an
natural numbers, and let lim
= 0. Then the following conditions are
n→∞ an+1
equivalent:
1. A(a) ⊆ A(b),
2. there exists a bounded matrix z such that b =∗ z · a,
∞
∑
i.e., (∃M ) (∀m > M ) bm =
zm,n an .
n=0
While the proof of implication 2.⇒1. is trivial, the proof of the opposite
implication is rather difficult and splits up to several lemmas. For the details
we refer the reader to [6]. Let us note that not much is known about the
an
> 0.
case lim sup
n→∞ an+1
4. A problem of perfect permitted sets
The notion of permitted sets was introduced by J. Arbault [1].
Definition 4.1. A set X ⊆ T is permitted if X ∪ Y is an N-set for every
N-set Y .
J. Arbault and P. Erdős independently proved that very countable set is
permitted (see [1]).
The problem of the existence of perfect permitted sets has an interesting
history. In 1952, J. Arbault in his dissertation [1] presented an example of
a perfect permitted set. However, in 1961, N. K. Bari [2] found a gap in
Arbault’s proof and stated the existence of a perfect permitted set as an
Thin sets of reals related to trigonometric series
5
open problem. In 1969, J. Lafontaine published a paper [10] proving that
there is no perfect permitted sets, but his proof seems to contain a gap, too.
During 1995–2000, several consistently uncountable examples of permitted sets were constructed by L. Bukovský, M. Repický, T. Bartoszyński,
I. Recław, M. Scheepers (see, e.g., [3]). It seemed that to prove the existence of an uncountable permitted set, one needs to use some additional
set-theoretic assumptions. In this situation, L. Bukovský conjectured that
every permitted set is perfectly meager, i.e., it is of first category relatively
to any perfect set. This fact was proved first for the family of Arbault sets,
later also for other families of trigonometric thin sets, including N-sets.
The notion of permitted sets can be defined for arbitrary family of sets.
Definition 4.2. Let F be a family of sets. A set X is called F-permitted
if X ∪ Y ∈ F for every Y ∈ F .
Related is the following notion.
Definition 4.3. Let F be a family of subsets of a group G. A set X ⊆ G
is called F-additive if X + Y ∈ F for every Y ∈ F .
Let Perm(F) denote the family of all F-permitted sets. If the family
F is hereditary then Perm(F) is an ideal. If F is hereditary and has a
base consisting of subgroups of G then a set X ⊆ G is F-permitted iff it is
F-additive.
Definition 4.4. A subset Y of a topological space X is called perfectly
meager if it is meager relatively to any perfect set, i.e., for any perfect set
P ⊆ X, P ∩ Y is meager (i.e., of first category) in topological space P .
We have two different proofs of that fact every A-permitted set is perfectly
meager. First one uses the characterization of inclusions between Arbault
sets (Theorem 3.3), see [6]. Another one uses the following strengthening of
a theorem of P. Erdős, K. Kunen, and R. D. Mauldin.
Theorem 4.5 (P. Erdős, K. Kunen, R. D. Mauldin, [8]). For any perfect
set P ⊆ R there exists a perfect set M having Lebesgue measure zero such
that P + M = R.
Theorem 4.6 (P. Eliaš, [7]). For any perfect set P ⊆ T there exists a
Dirichlet set D such that P + D = T.
The proof is based on the use of Kronecker’s theorem from approximation
theory. We immediately obtain the following.
Corollary 4.7. Let F be a family of subsets of T such that D ⊆ F and
T∈
/ F . Then there is no perfect F-additive set.
6
Peter Eliaš
Corollary 4.8. Let F be a hereditary family such that D ⊆ F and F
has a base consisting of proper subgroups of T. Then there is no perfect
F-permitted set.
Since D ⊆ N and T ∈
/ N , we have shown no perfect set is permitted.
The following notion was introduced by A. Nowik, M. Scheepers, and
T. Weiss [11] under the name “AFC′ -sets”. Later on, the authors used the
term “perfectly meager in transitive sense”. I propose to use a shorter, more
comfortable name “transitively meager sets”.2
Definition 4.9. A subset X of a topological group G is called transitively
meager if for every perfect set P ⊆ G there exists an Fσ -set F ⊇ X such
that for every y ∈ T, P ∩ (X + y) is meager in the relative topology of P .
Every transitively meager set of reals is perfectly meager. Under continuum hypothesis, one can construct a perfectly meager set which is not
transitively meager (see [11]).
By a slight modification of a proof of Theorem 4.6 one can prove the
following theorem.
Theorem 4.10. For any perfect set P ⊆ T there exists a pseudo-Dirichlet
set D such that for every y ∈ T, P ∩ (D + y) is dense in P .
The next theorem then follows.
Theorem 4.11 (P. Eliaš, [7]). Let F be a hereditary family of subsets of T
such that pD ⊆ F and for every E ∈ F there exists an Fσ -set F ⊇ E
satisfying E + F ̸= T. Then every F-permitted set is transitively meager.
As a corollary we obtain that if F is any of the families N , N0 , pD, A
then every F-permitted set is transitively meager.
5. Families generated by analytic subgroups
It is natural to ask, which families F of subsets of T satisfy the condition
from Theorem 4.11: every E ∈ F can be covered by an Fσ -set F such that
E + F ̸= T. Families N , N0 , pD, A fulfil this condition. In Corollary 5.3
we will show that this condition is satisfied by any sufficiently nice family.
A set in a Polish space is called analytic if it is a continuous image of a
Borel set.
In 1998, M. Laczkovich [9] proved the following theorem.
Theorem 5.1. Let E be a proper analytic subgroup of R. Then there exists
an Fσ -set F ⊇ E such that F has Lebesgue measure zero.
2Actually, I heard this term from O. Zindulka.
Thin sets of reals related to trigonometric series
7
We have proved the following strengthening of Laczkovich’s theorem.
Theorem 5.2 (P. Eliaš, [7]). Let E be a proper analytic subgroup of T.
Then there exists an Fσ -set F ⊇ E such that E + F has Lebesgue measure
zero.
Corollary 5.3. Let F be a hereditary family having a base consisting of
proper analytic subgroups of T. Then every E ∈ F can be covered by an
Fσ -set F such that E + F ̸= T.
Since every pseudo-Dirichlet set is contained in a group generated by a
Dirichlet set, from Theorem 4.11 we obtain the following.
Theorem 5.4. Let F be a hereditary family having a base consisting of
proper analytic subgroups of T and containing all Dirichlet sets. Then every
F-permitted set is transitively meager.
References
[1] Arbault J., Sur l’ensemble de convergence absolue d’une série trigonométrique, Bull.
Soc. Math. France 80 (1952), 243–317.
[2] Bari N.K., Trigonometric series, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow 1961,
936 pp.
[3] Bukovský L., Kholshchevnikova N.N., Repický M., Thin sets of harmonic analysis
and infinite combinatorics, Real Anal. Exchange 20 (1994/95), 454–509.
[4] Eliaš P., Covering for category and trigonometric thin sets, Proc. Amer. Math. Soc.
131 (2003), 3241–3249.
[5] Eliaš P., On inclusions between Arbault sets, Acta Univ. Carolin. Math. Phys. 44
(2003), 65–72.
[6] Eliaš P., Arbault permitted sets are perfectly meager, Tatra Mt. Math. Publ. 30 (2005),
135–148.
[7] Eliaš P., Dirichlet sets and Erdős-Kunen-Mauldin theorem, to appear. Preprint available at http://arxiv.org/abs/0712.2112.
[8] Erdős P., Kunen K., Mauldin R.D., Some additive properties of sets of real numbers,
Fund. Math. 113 (1981), 187–199.
[9] Laczkovich M., Analytic subgroups of the reals, Proc. Amer. Math. Soc 126 (1998),
1783–1790.
[10] Lafontaine J., Réunions d’ensembles de convergence absolue, Mém. Soc. Math. France
19 (1969), 21–25.
[11] Nowik A., Scheepers M., Weiss T., The algebraic sum of sets of real numbers with
strong measure zero sets, J. Symbolic Logic 63 (1998), 301–324.
[12] Salem R., The absolute convergence of trigonometrical series, Duke Math. J. 8 (1941),
317–334.
Peter Eliaš
Mathematical Institute,
Slovak Academy of Sciences,
Grešákova 6, 04001 Košice, Slovakia
Email address: elias@saske.sk