Howtorecognizeatrue Σ set

158 (1998)

How to recognize a true

Σ

by

Etienne M a t h e r o n (Talence)

Abstract.LetX be a Polish space, and let (Ap)p∈ω be a sequence ofG_δ hereditary subsets ofK(X) (the space of compact subsets ofX). We give a general criterion which allows one to decide whetherS

p∈ωApis a trueΣ⁰₃subset ofK(X). We apply this criterion to show that several natural families of thin sets from harmonic analysis are trueΣ⁰₃.

1. Introduction.In this paper, we are interested in a particular instance of the following problem: letX be a separable metric space, and denote by Σ⁰_ξ(X) (resp. Π⁰_ξ(X)) the additive (resp. multiplicative) Borel classes of X. The problem is to find some simple criterion allowing one to decide whether a givenΣ⁰_ξ setA ⊆ X is a “true”Σ⁰_ξ, that is, aΣ⁰_ξ set which is not Π⁰_ξ.

As a matter of fact, we will limit ourselves to the third level of the Borel hierarchy (G_δσandF_σδsets). Moreover, since the examples we have in mind are ideals of compact sets coming from harmonic analysis, we will concen- trate on proving criteria of “true-Σ⁰₃-ness” for ideals of compact subsets of some Polish space X. We denote by K(X) the space of all compact subsets of X, equipped with its natural (Polish) topology, generated by the sets {K ∈ K(X) :K∩V 6=∅} and {K ∈ K(X) : K ⊆V}, where V is an open subset ofX. For any subsetM ofX, we letK(M) ={K ∈ K(X) :K ⊆M}.

In this particular setting, it turns out that the simplest nontriviality condition is enough to ensure true-Σ⁰₃-ness. To be precise, let (A_p)_p∈ω be a sequence of dense Gδ hereditary subsets of K(X), and let A =S

p∈ωAp. Assume that A is an ideal of K(X), and that the union is “nontrivial” in the following sense: for each nonempty open set V ⊆X and for eachp∈ω, Ap∩ K(V) is a proper subset of A ∩ K(V). Then one can conclude that A is a true Σ⁰₃ set.

We prove this in the first part of the paper together with some related results. We apply these results in the second part to show that quite a lot of

1991Mathematics Subject Classification: 43A46, 04A15.

[181]

natural families of thin sets from harmonic analysis happen to be true Σ⁰₃ (a rather curious descriptive phenomenon). In particular, we show that ifG is a second-countable nondiscrete locally compact abelian group, then the familyHof compact Helson subsets ofGis a trueΣ⁰₃. The same result holds within anyM₀set, andH is also a trueΣ⁰₃inside the countable sets. In the case of the circle group, we already proved these results in [M]. However, the proofs given there were somewhat obscured by an immoderate use of constructions which are very classical in harmonic analysis, but still rather technical. In the present paper, we actually use very little harmonic analysis.

2. General results. In this section, X is a Polish space, K(X) is the space of compact subsets ofX, and Kω(X) is the family of countable com- pact subsets of X.

Definition 1. Let A be a subset ofK(X).

(a)Ais said to be hereditary if it is downward closed under inclusion.

(b)Ais anidealofK(X) if it is hereditary and stable under finite unions.

(c) IfAis hereditary, we say thatAis abig subset ofK(X) if it contains a denseGδ hereditary subset of K(X).

It is quite possible that any comeager hereditary subset of K(X) is big, but we are unable to prove it.

Definition 2. Let M1 and M2 be two subsets of K(X). We say that M₁ is nowhere contained in M₂ if for each nonempty open set V ⊆ X, M₁∩ K(V) is not contained in M₂.

We can now state the main results of this section.

TheoremA.Let (A_p)_p∈ω be a sequence of nonempty hereditary subsets of K(X), and let A=S

p∈ωA_p. Assume thatA is a big ideal ofK(X).

(a)If A is nowhere contained in any A_p,then A is notΠ⁰₃ in K(X).

(b) If the perfect sets in A are nowhere contained in any Ap, then the family of perfect sets in A is notΠ⁰₃ in K(X).

(c) If the finite sets in A are nowhere contained in any Ap, then A ∩ K_ω(X) is not relatively Π⁰₃ in K_ω(X).

Theorem A follows immediately from a more precise and less readable result, Theorem B below, which we will state after a few definitions.

In the sequel, we will use the notationsBandB₁for the following families of compact sets: eitherB =B1 =K(X), or B =B1 = the family of perfect compact subsets of X, or else B = {∅} ∪ {{x} : x ∈ X} and B₁ = {K ∈ K(X) :K is of the form{x} ∪ {xn:n∈ω}, where (xn)⊆X andxn→x}.

Notice that in each case, ∅ ∈ B and Bis aG_δ subset of K(X).

IfMis a subset ofK(X), we denote byM^f the family of compact subsets of X which are finite unions of elements ofM.

We denote by2^ω the Cantor space of all infinite 0-1 sequences, endowed with its usual (compact, metrizable) topology.

If α ∈ 2^ω and p ∈ ω, we define α_p ∈ 2^ω by α_p(q) = α(hp, qi), where (p, q)7→ hp, qi is any fixed bijection from ω² onto ω.

Finally, let Q= {α ∈ 2^ω :∃k ∀n≥ k α(n) = 0}, and W = {α ∈ 2^ω :

∃p α_p6∈Q}. It is well known thatW is a true Σ⁰₃subset of 2^ω (see [Ke2]).

Our slightly more precise version of Theorem A now reads as follows.

To deduce Theorem A from it, one just has to take B=K(X) in case (a), B = the perfect subsets of X in case (b), and B ={∅} ∪ {{x} :x ∈ X} in case (c).

TheoremB.Let(Ap)p∈ω be a sequence of(nonempty)hereditary subsets of K(X), and let A be any big subset of K(X). Assume that (A ∩ B)^f is nowhere contained in anyA_p. Then there exists a continuous mapα7→E(α) from 2^ω into K(X) such that:

• For eachα∈2^ω, E(α)∈ B1.

• If α∈W,then E(α)∈ A^f.

• If α6∈W,then E(α)6∈S

p∈ωA_p.

In particular, there is no Π⁰₃ set M ⊆ K(X) such that A^f∩ B₁⊆ M ∩ B₁⊆S

p∈ωA_p.

As an immediate consequence, we get a kind of “Baire category theorem”

for bigΠ⁰₃ ideals:

Corollary.LetA ⊆ K(X) be a bigΠ⁰₃ ideal. If (A_p)_p∈ω is a sequence of hereditary subsets of K(X) such that A ⊆S

p∈ωA_p, then there exist an integer p and a nonempty open set V ⊆X such that A ∩ K(V)⊆ Ap.

Some simple remarks may help to justify the hypotheses of Theorem B.

Assume thatX is perfect.

1) IfAis not big, the result is not true. For example, letAbe the ideal of finite sets and Ap = {K ∈ K(X) : card(K) ≤ p} (p ∈ ω); then A is nowhere contained in anyA_p, but it is obviously an F_σ set.

2) One cannot drop the hypothesis that A is an ideal. For example, let D = {x_n :n ∈ ω} be a countable dense subset of X, and let G =X\D.

Define A =K(G)∪S

n∈ωK({xn}) and Ap =K(G)∪S

n≤pK({xn}). Then Ais big, theA_p’s are hereditary and Ais nowhere contained in anyA_p; yet A isΠ⁰₃(it is the union of a G_δ and a countable set).

3) Finally, one cannot remove the hereditarity assumption on theA_p’s.

For example, let{Kp :p ∈ω} be any countable dense subset of K(X) (the K_p’s being pairwise distinct), and let A_p =K(X)\ {K_n:n > p}.

In the proof of Theorem B, we will make use of the following two lemmas.

The first one is easy; the second one is proved by applying the Baire category theorem in 2^ω (identified withP(ω)).

Lemma 1. The map φ : K(X)× K(X) → K(X) defined by φ(K, L) = K∪L is (continuous and) open.

Lemma2 ([Ke1]).Let G ⊆ K(X) beG_δ,F ∈ K(X), and let(F_m)_m∈ω be a sequence converging to F inK(X). Assume that for each finite set I ⊆ω, the set F∪S

m∈IF_m belongs to G. Then the compact set F ∪S

m∈ωF_m is the union of two elements of G.

Next, we introduce some notation.

Recall that we denote by hp, qi the image of a pair (p, q) under some fixed bijection fromω²onto ω. The image of an integernunder the inverse map will be denoted by ((n)₀,(n)₁).

Let 2^<ω be the set of all finite 0-1 sequences (including the empty se- quence). We write |s|for the length of a sequences∈2^<ω. Ifs∈2^<ω (and s 6= ∅), we denote by s⁰ the immediate predecessor of s in the extension ordering.

Ifα∈2^ω and n∈ω, we denote byα_dn the length-ninitial segment ofα;

thus, if n≥1, then α_dn = (α(0), . . . , α(n−1)).

Next, we define inductively a sequence (θ_p)_p∈ω of functions from 2^<ω intoω∪ {+∞}in the following way:

(0) θp(∅) = +∞ for all p∈ω.

(i) If|s|=n+ 1 and (n)0> p, thenθp(s) =θp(s⁰).

(ii) If |s|=n+ 1, (n)₀≤p and s(n) = 0, then θ_p(s) =

θp(s⁰) if θp(s⁰)<+∞, n ifθ_p(s⁰) = +∞.

(iii) If|s|=n+ 1, (n)0≤p and s(n) = 1, thenθp(s) = +∞.

In other words, if we defineA_p(s) ={m <|s|: (m)₀≤p}, thenθ_p(s) = min{m ∈ Ap(s) : ∀m⁰ ∈ Ap(s), m⁰ ≥m, s(m⁰) = 0} (with the convention that min(∅) = +∞). Thus, if we denote bys_p:A_p(s)→ {0,1}the restriction of s to A_p(s), then θ_p(s) indicates the beginning of the longest “cofinal”

0-segment insp.

Finally, we define another sequence of functions from 2^<ω into ω ∪ {+∞}by

mp(s) = max(hp,0i, θp(s)).

The following facts will be useful later.

Claim 1.Let p∈ω and α∈2^ω. Assume that αl ∈Q for all l≤p.

(a)The sequence(m_p(α_dn))_n∈ω is eventually finite-valued and constant.

(b)If we let Mp= limn→∞mp(α_dn), then α(Mp) = 0 and

∀n≥M_p m_p(α_dn+1) =M_p and ((n)₀> por α(n) = 0).

P r o o f. Since for eachs∈2^<ω,m_p(S) ≥θ_p(s) and the first coordinate ofm_p(s) is≤p, we may content ourselves with proving (a) and (b) withθ_p in place of mp.

By definition ofQ, there is a smallest integerN with the following prop- erties:

(N)₀≤p, ∀n≥N (n)₀≤p⇒α(n) = 0.

The claim will be proved if we can show thatθ_p(α_dn+1) =N for alln≥N. (i) First, θp(α_dn+1) =θp(α_dN+1) for every integer n ≥N. This follows by induction from the definition of the functionθ_p: by the choice ofN, we have α(N) = 0 and (N)0≤p, hence θp(α_dN+1)<+∞; and ifn > N, then either (n)₀> p orα(n) = 0, soθ_p(α_dn+1) =θ_p(α_dn) for alln > N.

(ii) By (i), it is now enough to check thatθ_p(α_dN+1) =N. Let N₁=

the greatestn < N such that (n)0≤p if there is any,

−1 if there is no suchn.

By the choice of N, we have α(N₁) = 1 if N₁ ≥ 0; thus, in both cases θp(α_dN1+1) = +∞. Now, by the choice of N1, we also have (n)0> p for all nsuch N₁< n < N. This implies that θ_p(α_dN) =θ_p(α_dN1+1) = +∞. Thus θ_p(α_dN+1) =N.

Proof of Theorem B. The result is trivial if X is not perfect (one just has to let E(α) ≡ {x₀}, where x₀ is an isolated point of X). Hence, from now on,X will be perfect.

For simplicity, we assume first that X is compact. We fix some metric compatible with the topology of X and we choose a complete metric δ for B, which is possible since B is aG_δ subset of K(X).

Since each A_p is hereditary, the hypotheses of Theorem B remain un- changed if we replace Ap by S

l≤pAl. Thus, we may assume that the se- quence (A_p)_p∈ω is nondecreasing.

Finally, let G be a dense G_δ hereditary subset of K(X) contained in A.

We can write G =T

n∈ωUⁿ, where (Uⁿ)n∈ω is a nonincreasing sequence of hereditary open subsets ofK(X) (if (Wⁿ)_n∈ω is any nonincreasing sequence of open sets such that G = T

n∈ωWⁿ, let Uⁿ = {K ∈ K(X) : ∀L ⊆ K L∈ Wⁿ}).

Claim 2.For each positive integer N, the set {(x, K₁, . . . , K_N) ∈ X× B^N :{x} ∪S_N

i=1Ki∈ G} is dense inX× B^N.

P r o o f. By Lemma 1, the set{(K₀, K₁, . . . , K_N)∈ K(X)^N+1:S_N

i=0K_i

∈ G} is a dense Gδ subset of K(X)^N+1; and since G is hereditary, this implies that the set {(x, K₁, . . . , K_N) ∈X× K(X)^N :{x} ∪S_N

i=1K_i ∈ G}

is a dense Gδ subset of X× K(X)^N. Thus, the claim is true if B=K(X).

If B is the family of perfect sets, which is comeager in K(X) because X is perfect, the claim follows from the Baire category theorem. Finally, if B={∅} ∪ {{x}:x∈X}, we use again the fact that G is hereditary.

Now, we shall construct inductively a sequence (j_m)_m∈ω of positive inte- gers and, for each s∈2^<ω, a compact setE(s)⊆X and a nonempty open set V(s)⊆X.

For s 6= ∅, E(s) will be written as E(s) = S

m<|s|E^m(s), where each E^m(s) is compact and of the form E^m(s) =S_jm

j=1E_j^m(s) (E_j^m(s) compact).

We also construct (for s ∈ 2^<ω \ {∅}, 0 ≤ m < |s|, and 1 ≤ j ≤ j_m) nonempty open sets V_j^m(s)⊆X, and we letV^m(s) =S_jm

j=1V_j^m(s).

The closure of any set A involved in the construction will be denoted by A.

The following requirements have to be fulfilled (to avoid typographic heaviness, we have omitted more often than not obvious information like “if

|s| ≥1”, “m <|s|” or “j≤j_m”).

(1) E^m_j (s)∈ A ∩ B and E^m(s)6=∅.

(2)





E_j^m(s)⊆V_j^m(s)⊆V_j^m(s)⊆V_j^m(s⁰),

theV^m(s) are pairwise disjoint and disjoint fromV(s), V(s)⊆V(s⁰), Vⁿ(s)⊆V(s⁰) if|s|=n+ 1.

(3) δ(E_j^m(s), E_j^m(s⁰))<2^−|s|, diamV(s)<2^−|s|.

(4) If |s|=n+ 1, then E_j^mp(s)(s) =E_j^mp(s)(s⁰) for each p <(n)₀ (notice that mp(s) =mp(s⁰) here, becausep <(n)0).

(5) If |s|=n+ 1 and s(n) = 0, thenE_j^m(s) =E_j^m(s⁰) for all m <|s⁰|.

(6) If |s|=n+ 1 and s(n) = 0, thenEⁿ(s)6∈ An. (7) If |s|=n+ 1 and s(n) = 1, then

V(s)∪[

{V^m(s) :m <|s|, ∀p <(n)₀m6=m_p(s)} ∈ U^|s|. To begin the construction, we choose a nonempty open setV(∅)⊆X of diameter<1, and we let E(∅) =∅.

Assume that the sets E(t) and V(t) have been constructed for all se- quences t of length ≤n. We have to define the positive integer j_n and the setsV(s),E_j^m(s), V_j^m(s) (0≤m ≤n, 1≤j ≤j_m) for every sequence sof lengthn+ 1.

(a) First, for each sequencetof lengthn, we choose two nonempty open setsW1(t), W2(t) ⊆V(t) with W1(t)∩W2(t) = ∅. This is possible because X is perfect.

(b) Next, we definejn and the sets E_j^m(s) andV_j^m(s) for all sequences sof length n+ 1 such that s(n) = 0.

(i) By (5), there is nothing to do for an integerm < n.

(ii) LetS₀={s∈2^<ω :|s|=n+ 1, s(n) = 0}. Since (A ∩ B)^f is nowhere contained inAn, we can find a positive integerjnand, for alls∈S0, compact setsE₁ⁿ(s), . . . , E_jⁿ_n(s)⊆W₁(s⁰) such that eachE_jⁿ(s) belongs toA ∩ B, but S_jn

j=1E_jⁿ(s) 6∈ A_n. Notice that we can choose the same integer j_n for all sequencessbecause, since∅ ∈ A ∩ B, we may always add the empty set (as many times as necessary) to the setsE_jⁿ(s).

At this point, (4), (5) and (6) are satisfied, as well as (1) for s ∈ S₀ (Eⁿ(s) is nonempty because ∅ ∈ An). Then we can choose for each s∈S0

nonempty open sets V(s)⊆W₂(s⁰) andV_j^m(s)⊇E_j^m(s) in order to get (2) and (3).

(c) Now, lets be a sequence of lengthn+ 1 such that s(n) = 1.

(i) By (4), we must let E_j^mp(s)(s) =E_j^mp(s)(s⁰) for all the integers p <

(n)₀such that m_p(s) =m_p(s⁰)<|s⁰|.

(ii) Let I(s) ={m <|s|:∀p <(n)0 m6=mp(s)} and N =P

m∈I(s) jm. By Claim 2, the set {(x, K₁, . . . , K_N) ∈ X× B^N : {x} ∪S_N

i=1K_i ∈ G} is dense inX× B^N. Therefore, we can find a pointx(s)∈X and compact sets E_j^m(s)∈ B (m∈I(s), 1≤j≤jm) such that

(∗)









δ(E_j^m(s), E_j^m(s⁰))<2⁻ⁿ⁻¹and E_j^m(s)⊆V_j^m(s⁰) if m < n, E_jⁿ(s)⊆W1(s⁰) and E_jⁿ(s)6=∅,

x(s)∈W₂(s⁰), {x(s)} ∪S

{E_j^m(s) :m∈I(s), 1≤j ≤j_m} ∈ G.

We can also ensure that E_j^m(s) 6= ∅ whenever E_j^m(s⁰) 6= ∅, because ∅ is an isolated point in K(X). Moreover, since G is hereditary (and contained inA), the last condition implies that eachE_j^m(s) belongs to A; hence (1) is true for m∈I(s) (of course, it was also true form6∈I(s)).

(iii) It is now easy to choose open setsV(s)3x(s) and V_j^m(s)⊇E_j^m(s) in order to get (2), (3) and (7).

This concludes the inductive step.

It follows from (1) and (3) that if m ∈ ω and j ≤ j_m are fixed, then for any α ∈ 2^ω, the sequence (E_j^m(α_dn))n>m converges to a compact set E_j^m(α)∈ B.

For each α ∈2^ω and each m ∈ω, let E^m(α) =S

j≤jmE_j^m(α). By (1), all the E^m(α)’s are nonempty (because ∅ is isolated in K(X)). Moreover, conditions (2) and (3) imply that the sequence (E^m(α))_m∈ω converges in K(X) to the singleton{xα}=T

n∈ωV(α_dn).

Thus, the set E(α) = {x_α} ∪S

m∈ωE^m(α) is compact, and in fact it belongs to B1. Furthermore, it follows from (2) and (3) that the mapα 7→

E(α) is continuous.

Claim 3.Let α ∈2^ω and p∈ω. Assume that αl ∈Q for alll≤p,and let M_p= lim_n→∞ m_p(α_dn). Then E^Mp(α) =E^Mp(α_dMp+1).

P r o o f. By Claim 1, the integerMp is well defined. Moreover, we know that for eachn≥M_p,m_p(α_dn+1) =M_p, and either (n)₀> p orα(n) = 0.

This implies thatE^Mp(α_dn+1) =E^Mp(α_dn) for eachn > M_p. Indeed, we can use (4) if (n)₀> pand (5) ifα(n) = 0. Thus E^Mp(α) =E^Mp(α_dMp+1).

Let us now fixα∈2^ω.

Case 1.Assumeα6∈W. By Claims 1 and 3, all sequences (m_p(α_dn))_n∈ω are eventually constant, and if we letM_p= lim_n→∞ m_p(α_dn) (p∈ω), then α(Mp) = 0 and E^Mp(α) = E^Mp(α_dMp+1). Hence, by (6), E^Mp(α) 6∈ AM_p. Since each A_Mp is hereditary, this implies that E(α) 6∈ S

p∈ωA_Mp. Now M_p≥ hp,0ifor allp, hence lim_p→∞ M_p= +∞(this was the reason for using the functions mp rather than the θp’s), and consequentlyE(α)6∈S

p∈ωAp. Case 2.Assume α∈W. We have to show that E(α)∈ A^f.

Letp₀ be the smallest integer such that α_p6∈Q. For eachp < p₀, let as usual Mp= limn→∞ mp(α_dn) (which is well defined by Claim 1) and let

E₁(α) = [

p<p₀

E^Mp(α), E₂(α) ={x_α} ∪[

{E^m(α) :m6=M_p for all p < p₀}.

Since E(α) = E1(α)∪E2(α), it is enough to check that E1(α) ∈ A^f and E₂(α)∈ G^f.

(i) By the choice ofp0,αp∈Qfor each p < p0. Hence, by Claim 3 and condition (1), E₁(α) =S

p<p0E^Mp(α_dMp+1)∈ A^f.

(ii) Let I(α) = {m ∈ ω : ∀p < p₀ m 6= M_p}. It follows from (7) that V(α_dn+1)∪S

{V^m(α_dn+1) : m ∈ I(α), m < n+ 1} ∈ Uⁿ for each integer n > max{M_p : p < p₀} such that (n)₀ = p₀ and α(n) = 1. Since there are infinitely many such n’s (because αp₀ 6∈ Q) and each open set Uⁿ is hereditary, this implies (by (2)) that for any finite set I ⊆ I(α), {x_α} ∪S

m∈IE^m(α)∈ G. Thus, from Lemma 2 we getE₂(α)∈ G^f.

The proof of Theorem B is now complete when X is assumed to be compact.

When X is not compact, we may always view it as a dense Gδ subset of some compact metric space X. Writee X = T

n∈ωW_n, where the W_n’s are open subsets of X. Then we can perform our construction ine Xe and moreover, we can easily ensure at each step that the open sets V_j^m(s) and V(s) are all contained inW_|s|. Hence, in the end,E(α)⊆Xfor eachα∈2^ω. This concludes the whole proof.

3. Applications. In this section, G is a second-countable nondiscrete locally compact abelian group, with dual group Γ. We denote by C₀(G), M(G), A(G) and P M(G) respectively: the space of continuous complex- valued functions on G vanishing at infinity, the space of finite (complex) measures on G, the Fourier transform of the convolution algebra L¹(Γ), and the space of pseudomeasures onG (the dual space ofA(G)).

Let q(G) = sup{n∈ ω : every neighbourhood of 0_G contains elements of order ≥n}. We define Gq = {x ∈G :x is of order ≤q(G)}, and Tq = {z∈T:z^q(G) = 1}(with the convention thatz^∞ = 1 for anyz∈T). Notice thatG_q is a clopen subgroup ofG, by definition ofq(G).

Definition 1. Let K be a compact subset ofG.

1)K is said to be aHelson set if every continuous function onK can be extended to a function inA(G).

2)K is said to bewithout true pseudomeasure(for short,W T P) if every pseudomeasure supported by K is actually a measure.

3) K is said to be independent if there is no nontrivial relation of the form P_n

i=1m_ix_i = 0, where m₁, . . . , m_n ∈ Z and x₁, . . . , x_n ∈ K (that is, Pmixi = 0 ⇒ ∀i mi, xi = 0; when G = T, this is not exactly the usual definition).

4)K is said to be aK_q set if it is totally disconnected, all its elements have orderq(G), and the restrictions of characters ofGare uniformly dense inC(K,Tq), the space of continuous functions fromKintoTq(whenq(G) = +∞,K_q sets are usually calledKronecker sets).

5)K is said to be aU₀⁰ set if there is some constant csuch that

∀µ∈M+(K) kµkP M ≤c lim

γ→∞|bµ(γ)|.

It is clear thatH(the family of Helson subsets ofG),W T P,K_q andU₀⁰ are hereditary subsets of K(G) (for Kq sets, this is because they are assumed to be totally disconnected).

There are natural constants associated with a given Helson or U₀⁰ set.

Namely, for eachK ∈ K(G), define η₀(K) = inf{ lim

γ→∞|bµ(γ)|/kµk_{P M} :µ∈M₊(K), µ6= 0}, α(K) = inf{kµk_{P M}/kµk_M :µ∈M(K), µ6= 0}.

Then (by definition) K ∈ U₀⁰ ⇔ η0(K) > 0 and (by standard functional- analytic arguments) K ∈ H ⇔α(K)>0. The number α(K) is theHelson constant ofK. There is also a “W T P constant”, whose definition should be reasonably clear.

Lemma 1.K_q is aG_δ subset of K(G), and H,W T P andU₀⁰ are Σ⁰₃.

The proofs are standard complexity calculations. For Helson sets, for example, the main point is that for each positive ε, the set H_ε = {K ∈ K(X) :α(K)≥ε}is G_δ.

Definition 2. Let E be a closed subset ofG.

(a)E is said to be aU0 set(or a set ofextended uniqueness) ifµ(E) = 0 for every positive measure onGwhose Fourier transform vanishes at infinity.

(b) E is an M₀ set if E 6∈ U₀, and an M₀^p set if E∩V ∈M₀ for each open setV ⊆Gsuch that E∩V 6=∅.

Definition 3. Let p be a positive integer. By a net of length p, we mean any set of cardinality 2^p of the form{a+P_p

i=1εili:εi= 0,1}, where a, l₁, . . . , l_p are fixed elements of G.

We shall use the following results. Almost all the proofs can be found in [GMG], and a lot of them in [KL] (see also [LP]).

1)W T P ⊆ H ⊆U₀⁰ ⊆U₀.

2) H, W T P and U₀⁰ are translation-invariant ideals of K(G); U₀ is a translation-invariant σ-ideal of closed sets.

3) Finite sets are W T P. A finite set is a Kq set if and only if it is independent and all its elements have order q(G).

4)K_q sets are Helson;in fact, α_q = inf{α(K) :K ∈K_q} is >0.

5)If F ⊆G is a p-net,thenα(F)≤(√

2)^−p. Hence,if a compact setK contains arbitrarily long nets,thenK is not Helson.

Before applying the results of Section 2, we prove some general facts about K_q sets.

For each integer m such that 0 < m < q(G), we let N_m = {x ∈ G : mx= 0}. The Nm’s are closed subgroups of Gand, by definition of q(G), they are nowhere dense inG.

We denote byI theσ-ideal generated by all translates of theN_m’s, that is, the family of all subsets of G which can be covered by countably many translates of the N_m’s.

Finally, we say that a set E ⊆G isI-perfect if no nonempty relatively open subset of E belongs to I. By the Baire category theorem, every open subset of Gis I-perfect.

Lemma 2. Let F ⊆ G be a finite Kq set, and let A = {x ∈ G : x ∈ G_q and F∪ {x} 6∈K_q}. ThenA∈ I.

P r o o f. We know thatF∪ {x}is aKq set if and only ifxhas orderq(G) andF∪ {x}is independent. Moreover, ifx∈Ghas order≤q(G), it is easy to check that for eachm∈Z, there is an integer m⁰ such that|m⁰|< q(G) and mx=m⁰x.

Now, let Gp(F) be the subgroup of G generated by F. From the two preceding remarks, we easily deduce that A is contained in the set A⁰ de- fined by

x∈A⁰⇔ ∃m (0<|m|< q(G) and mx∈Gp(F)).

If 0 <|m| < q(G) and y ∈G, the set E_m,y ={x ∈G :mx =y} belongs to I. Since Gp(F) is countable, it follows that A⁰ ∈ I. This concludes the proof.

Theorem1.Let E⊆Gbe a closedI-perfect set contained inG_q. Then Kq∩ K(E) is dense in K(E). In fact, for any finite Kq set F ⊆G,the set G_F ={K∈ K(E) :K∪F ∈K_q} is a dense G_δ hereditary subset of K(E).

P r o o f. Since K_q is hereditary, the second statement implies the first.

So let us fix a finiteKq setF ⊆G.

It is clear thatG_F is G_δ and hereditary.

Now, let V1, . . . , Vk be nonempty open subsets of E. Since each Vi is I-perfect, we can apply Lemma 2 k times to get x₁, . . . , x_k ∈G such that x_i ∈V_i for all iand F∪ {x₁, . . . , x_k} ∈K_q. This shows thatG_F is dense in K(E).

In the circle group, Theorem 1 simply says that the Kronecker sets are dense in any perfect subset of T, which is a well known fact. Whenq(G)<

∞, simple examples show that even ifP ⊆Gis perfect and all its elements have order q(G), theK_q sets contained inP need not be dense inK(P).

Corollary. Let E ⊆ G be an I-perfect set. Then W T P ∩ K(E) is a big subset of K(E).

P r o o f. It is easy to check (using the Baire category theorem and the separability of G) that, given any nonempty closed set F ⊆G, there exist a pointa∈Gand an open setV such thatV ∩F 6=∅anda+F∩V ⊆G_q. Thus we may and do assume that E is contained in G_q (because W T P is translation-invariant and every open subset ofE is I-perfect).

Now, letα_q be the constant introduced above, and let G be the family of all compact sets K⊆E with the following property:

∀S∈B1(P M(K))∀f ∈A(G) |hS, fi| ≤αq sup{|f(x)|:x∈E}.

SinceA(G) is dense inC₀(G),G is contained inW T P. Moreover, using the separability ofA(G), one easily checks thatG is aG_δ subset ofK(E), which is obviously hereditary.

Finally, since finite sets are W T P, G contains every finite K_q subset of E; hence, by Theorem 1 (and the fact thatKq is hereditary),G is dense inK(E). This concludes the proof.

Now we turn to applications of the results of Section 2. We show first that W T P, H and U₀⁰ are true Σ⁰₃ within any M₀ set; then we prove that His trueΣ⁰₃ inside the countable sets.

Lemma 3.Every closed set in I is a U0 set.

P r o o f. For each integer m such that 0 < m < q(G), N_m is a closed but nonopen subgroup of G. By a result of V. Tardivel [T], this implies thatN_m∈U₀. SinceU₀ is a translation-invariantσ-ideal of closed sets, the lemma follows.

Lemma 4. Let E ⊆ G be a nonempty M₀^p set and for p ∈ ω, define A_p={K ∈ K(E) :η₀(K)≥2^−p}. Then the perfect WTP sets contained in E are nowhere contained in any A_p.

P r o o f. By Lemma 3,E is I-perfect. Hence, by Theorem 1 (Corollary), W T P ∩ K(E) is a big subset of K(E). Now, by a result of Kechris [Ke1], ifF is anyM₀^p set and G is any dense Gδ hereditary subset of K(F), then, for each integer N ≥1, there exist perfect sets K₁, . . . , K_N ∈ G such that η₀(K₁∪. . .∪K_N)<4/N. This proves the lemma.

LetP be the family of perfect compact subsets ofG.

Theorem 2.If E ⊆Gis an M₀ set, then there is no Π⁰₃ set such that P ∩W T P ∩ K(E) ⊆ M ⊆ U₀⁰. In particular, the families of perfect W T P sets,perfect Helson sets and perfect U₀⁰ sets contained in E are true Σ⁰₃ in K(E).

P r o o f. Since U0 is a σ-ideal of closed sets, every M0 set contains a nonemptyM₀^pset. Hence, Theorem 2 follows from Lemma 4 and Theorem B.

Lemma 5. Let p be a positive integer, and let V be a nonempty open subset of Gcontained in Gq. Then there is a finite setF ⊆G such that

• F ⊆V.

• F is a net of length p.

• Each element of F has order q(G).

P r o o f. By Theorem 1 applied to G_q, the set {(x₀, . . . , x_p) ∈ G^p+1_q : {x0, . . . , xp} ∈Kq}is dense inG^p+1_q . Now, for eachx= (x0, . . . , xp)∈G^p+1_q , let F(x) ={x₀+P_p

i=1ε_ix_i :ε_i = 0,1}. The mapF is clearly continuous, so the set{x∈G^p+1_q :F(x)⊆V, x_i6=x_j for all i6=j}is a nonempty open subset ofG^p+1_q . It follows that there exist pairwise distincta, l1, . . . , lp∈Gq

such thatF(a, l₁, . . . , l_p)⊆V and {a, l₁, . . . , l_p} is aK_q set.

It is then easy to see that F = F(a, l1, . . . , lp) has cardinality 2^p and that each element ofF has orderq(G). This proves the lemma.

Theorem 3. There exists a continuous map α 7→ E(α) from 2^ω into K(G) such that

• For eachα∈2^ω, E(α) is a convergent sequence.

• If α∈W,then E(α) is the union of finitely manyK_q sets.

• If α6∈W,then E(α) contains arbitrarily long nets.

In particular, the countable Helson sets form a trueΣ⁰₃ subset of K_ω(G).

P r o o f. By Theorem 1 and Lemma 5, we can apply Theorem B with X=Gq,G=Kq∩ K(X),B={∅} ∪ {{x}:x∈X} andAp={K∈ K(X) : K does not contain anyp-net}.

Notice thatH∩K_ω(G) is notΣ⁰₃inK(G). In fact, it is not even Borel. To see this, take an independentM0 set E ⊆G (aRudin set, see [LP]). Then K_ω(E) is not Borel in K(G), since E is uncountable. But every countable independent compact set is Helson (see [KL]). Hence H ∩ Kω(E) = Kω(E) is not Borel in K(G). The same example shows that we cannot “localize”

Theorem 3 within an arbitraryM₀ set.

To conclude this paper, we briefly discuss other examples of naturalΣ⁰₃ in harmonic analysis.

Very close to the U₀⁰ sets are the U⁰ sets (see [KL]) and the U₂⁰ sets of R. Lyons [Ly], which also form Σ⁰₃ subsets of K(T). In fact, one has the inclusions W T P ⊆U⁰ ⊆ U₂⁰ ⊆ U₀⁰, so (by Theorem 2) U⁰ and U₂⁰ are true Σ⁰₃. ForU⁰ sets, there is a more precise “local” result: ifE is any closed set of multiplicity, then U⁰∩ K(E) is a true Σ⁰₃ of K(E). This can be deduced from our criteria, using the family of Dirichlet sets rather than the family of Kq sets.

Other examples in the circle group are thep-Helson sets introduced by M. Gregory [G]. A compact setK ⊆Tis p-Helson (say K ∈ H_(p)) if every continuous function on K is the restriction of a continuous function on T whose Fourier series belongs to l^p(Z). Obviously, H₍₁₎=H,H_(p) ⊆ H_(p⁰₎ if p≤p⁰, and H₍₂₎=K(T).

It is shown in [G] that for p > 1, H_(p) is a σ-ideal of K(T), and that a compact set K is p-Helson if and only if for all µ ∈ M(K), kµkM = inf{kµ−λk_M+kλkb _lq :λ∈M_q(T)}, whereM_q(T) ={λ∈M(T) :λb∈l^q(Z)}

(and q = p/(p−1)). Using this, it is not hard to check that H_(p) is G_δ in K(T).

Now, letHebe the family of compact subsets of Twhich arep-Helson for some p ∈ ]1,2[. It follows from the preceding remarks that He is a big Σ⁰₃ ideal ofK(T). Moreover, it is also shown in [G] that for anyp <2,H_(p) is a proper subset ofH; and it is not difficult to deduce from the proof that thise is true in any open subset ofT. Thus Theorem A applies, and we conclude thatHe is a true Σ⁰₃ subset of K(T).

On the other hand, our criteria do not apply to the family ofH-sets of the circle group, which is also a trueΣ⁰₃(see [Li]), becauseHis not an ideal of K(T).

References

[GMG] C. C. G r a h a m and O. C. M c G e h e e,Essays in Commutative Harmonic Anal- ysis, Grundlehren Math. Wiss. 238, Springer, New York, 1979.

[G] M. B. G r e g o r y,p-Helson sets, 1< p <2, Israel J. Math. 12 (1972), 356–368.

[Ke1] A. S. K e c h r i s,Hereditary properties of the class of closed sets of uniqueness for trigonometric series, ibid. 73 (1991), 189–198.

[Ke2] —,Classical Descriptive Set Theory, Springer, New York, 1995.

[KL] A. S. K e c h r i s and A. L o u v e a u,Descriptive Set Theory and the Structure of Sets of Uniqueness, London Math. Soc. Lecture Note Ser. 128, Cambridge Univ.

Press, 1987.

[LP] L.-A. L i n d a h l and A. P o u l s e n, Thin Sets in Harmonic Analysis, Marcel Dekker, New York, 1971.

[Li] T. L i n t o n,TheH-sets of the unit circle are properlyG_δσ, Real Anal. Exchange 19 (1994), 203–211.

[Ly] R. L y o n s,A new type of sets of uniqueness, Duke Math. J. 57 (1988), 431–458.

[M] E. M a t h e r o n,The descriptive complexity of Helson sets, Illinois J. Math. 39 (1995), 608–625.

[T] V. T a r d i v e l,Ferm´es d’unicit´e dans les groupes ab´eliens localement compacts, Studia Math. 91 (1988), 1–15.

Universit´e Bordeaux 1 351, Cours de la lib´eration 33405 Talence Cedex, France

E-mail: matheron@math.u-bordeaux.fr

Received 15 April 1998