aa r X i v : . [ m a t h . L O ] D ec ON THE DESCRIPTIVE COMPLEXITY OF SALEM SETS
ALBERTO MARCONE AND MANLIO VALENTI
Abstract.
In this paper we study the notion of Salem set from the pointof view of descriptive set theory. We first work in the hyperspace K ([0 , ,
1] and show that the closed Salem sets form a Π -complete family. This is done by characterizing the complexity of the familyof sets having sufficiently large Hausdorff or Fourier dimension. We also showthat the complexity does not change if we increase the dimension of the ambientspace and work in K ([0 , d ). We then generalize the results by relaxing thecompactness of the ambient space, and show that the closed Salem sets arestill Π -complete when we endow F ( R d ) with the Fell topology. A similarresult holds also for the Vietoris topology. Contents
1. Introduction 12. Background 33. The complexity of closed Salem subsets of [0 ,
1] 74. The complexity of closed Salem subsets of [0 , d R d Introduction
The notion of Salem set arises naturally in the context of geometric measuretheory and the theory of fractal dimension. A set A ⊂ R d is called Salem iffdim H ( A ) = dim F ( A ), where dim H and dim F denote the Hausdorff and the Fourierdimension respectively.Hausdorff dimension is a fundamental notion in geometric measure theory andcan be found in almost every textbook in the field. It describes the “size” of aset by means of the diameter of open sets covering it. When working with Borelsubsets of R d , Frostman’s lemma characterizes the Hausdorff dimension of a set bymeans of the existence of finite Radon measures supported on the set with certainregularity properties (see Section 2 for details).This characterization establishes a close connection with the Fourier transformof a measure. Indeed, it can be shown that the decay of the Fourier transform of a(probability) measure supported on the set provides a lower bound for the Hausdorffdimension. This leads to the notion of Fourier dimension and hence to the one of Mathematics Subject Classification.
Primary: 03E15; Secondary: 28A75, 28A78, 03D32.
Key words and phrases.
Salem sets, Hausdorff dimension, Fourier dimension, Borel hierarchy.
Salem set. It is known that, for Borel subsets of R d , the Fourier dimension neverexceeds the Hausdorff dimension.The first non-trivial examples of Salem sets were based on random constructions([22, 14]). Later Kahane [13] modified the original construction by Salem to producean explicit Salem set of dimension α , for every α ∈ [0 , α ≥
0, the set E ( α )of α -well approximable numbers is a fractal with Hausdorff dimension 2 / (2 + α ).Kaufmann [15] improved the result by showing that there is a probability measuresupported on a subset of E ( α ) witnessing the fact that dim F ( E ( α )) ≥ / (2 + α ),which implies that E ( α ) is Salem (the reader is referred to [4] or [24] for detailedproofs of Kaufmann’s theorem).A classical example of a non-Salem set is Cantor middle-third set, which hasFourier dimension 0 and Hausdorff dimension log(2) / log(3). Similarly, every sym-metric Cantor set with dissection ratio 1 /n , with n >
1, is not Salem, as it hasnull Fourier dimension and Hausdorff dimension log(2) / log( n ) (see [20, Sec. 4.10]and [21, Thm. 8.1]). It can be proved that, for every 0 ≤ x ≤ y ≤ ,
1] with Fourier dimension x and Hausdorff dimension y ([19,Thm. 1.4]).There are not many explicit (i.e. non-random) examples of subset of R d whichare known to be Salem. As a corollary of a result of Gatesoupe [10], we know thatif A ⊂ R is a Salem set of dimension α then the set { x ∈ R d : | x | ∈ A } is Salemand has dimension d − α . Recently, using a higher-dimensional analogue of E ( α ), some explicit examples of Salem subsets of R ([11]) and R d ([9]) of arbitrarydimension have been constructed.In this paper we study the complexity, from the point of view of descriptive settheory, of the family { A ∈ F ( X ) : A ∈ S ( X ) } , where F ( X ) is the hyperspace ofclosed subsets of X , S ( X ) is the family of Salem subsets of X , and X is either[0 , , d or R d . In other words we study the complexity of the property “being aSalem set”, when we restrict our attention to closed sets. For the sake of readabilitywe write S c ( X ) := S ( X ) ∩ F ( X ) for the set of closed Salem subsets of X . Weshow that it is Borel and classify it in the Borel hierarchy.We summarize our results for X = [0 ,
1] in the following table. p < { A ∈ K ([0 , H ( A ) > p } Σ -complete p > { A ∈ K ([0 , H ( A ) ≥ p } Π -complete p < { A ∈ K ([0 , F ( A ) > p } Σ -complete p > { A ∈ K ([0 , F ( A ) ≥ p } Π -complete { A ∈ K ([0 , A ∈ S ([0 , } Π -completeThe complexities remain the same if we replace [0 ,
1] with any interval, with [0 , d or R d . In particular, the fact that the family of closed Salem subsets of [0 ,
1] is Π -complete answers a question asked by Slaman during the IMS Graduate SummerSchool in Logic, held in Singapore in 2018.Our results can be used to obtain the classifications of the functions computingthe dimensions of closed sets, both in the Baire hierarchy and in the effectivehierarchy defined via Weihrauch reducibility, in particular answering also a question N THE DESCRIPTIVE COMPLEXITY OF SALEM SETS 3 raised by Fouch´e ([5]) and Pauly. The details will be explored in a forthcomingpaper.
Acknowledgements.
The early investigations leading to this paper were motivatedby the above-mentioned question asked by Ted Slaman. The results of Section 3are indeed joint work with Ted Slaman and Jan Reimann.We would also like to thank Geoffrey Bentsen, Riccardo Camerlo, Kyle Ham-brook, Arno Pauly and Linda Brown Westrick for useful discussions and suggestionson the topics of the paper.Both author’s research was partially supported by the Italian PRIN 2017 Grant“Mathematical Logic: models, sets, computability”. Valenti’s participation to theIMS Graduate Summer School in Logic in 2018 was partially funded by the Institutefor Mathematical Sciences (National University of Singapore).2.
Background
For a general introduction to geometric measure theory the reader is referred to[8]. Here we introduce the notions and notations we will use throughout the rest ofthe paper.Let X be a separable metric space and let A ⊂ X . Let also diam( A ) denote thediameter of A . We say that a family { E i } i ∈ I is a δ -cover of A if A ⊂ S i ∈ I E i anddiam( E i ) ≤ δ for each i ∈ I . For every s ≥ δ ∈ (0 , + ∞ ] we define H sδ ( A ) := inf (X i ∈ I diam( E i ) s : { E i } i ∈ I is a δ -cover of A ) , H s ( A ) := lim δ → + H sδ ( A ) = sup δ> H sδ ( A ) . The function H s is called s -dimensional Hausdorff measure . The Hausdorff dimen-sion of A is defined asdim H ( A ) := sup { s ∈ [0 , + ∞ ) : H s ( A ) > } . As a consequence of Frostman’s lemma (see [20, Thm. 8.8, p. 112]), for every Borelsubset A of R d (with the Euclidean norm), the Hausdorff dimension of A coincideswith its capacitary dimension dim c ( A ), defined assup { s ∈ [0 , d ] : ( ∃ µ ∈ P ( A ))( ∃ c > ∀ x ∈ R d )( ∀ r >
0) ( µ ( B ( x, r )) ≤ cr s ) } , where P ( A ) is the set of Borel probability measures with support included in A and B ( x, r ) denotes the ball with center x and radius r . We notice that the Hausdorffdimension is countably stable (i.e. for every family { A i } i ∈ N we have dim H ( S i A i ) =sup i dim H ( A i ), see [20, p. 59]) and, for every α -H¨older continuous map f : R n → R m we have dim H ( f ( A )) ≤ α − dim H ( A ) (see [8, Prop. 3.3, p. 49]). In particular everybi-Lipschitz map preserves the Hausdorff dimension.For every probability measure µ on R d , we can define the Fourier transform of µ as the function b µ : R d → C defined as b µ ( ξ ) := Z R d e − i ξ · x dµ ( x )where ξ · x denotes the scalar product. We define the Fourier dimension of A ⊂ R d asdim F ( A ) := sup { s ∈ [0 , d ] : ( ∃ µ ∈ P ( A ))( ∃ c > ∀ x ∈ R d ) ( | b µ ( x ) | ≤ c | x | − s/ ) } . ALBERTO MARCONE AND MANLIO VALENTI
If we define dim F ( µ ) := sup { s ∈ [0 , d ] : ( ∃ c > ∀ x ∈ R d ) ( | b µ ( x ) | ≤ c | x | − s/ ) } then we have dim F ( A ) = sup { dim F ( µ ) : µ ∈ P ( A ) } . For background notions onthe Fourier transform the reader is referred to [23]. For its applications to geometricmeasure theory see [21].The Fourier dimension is not as stable as the Hausdorff dimension. Some stabil-ity properties of the Fourier dimension have been investigated in [6]. We underline,however, that the definition of Fourier dimension given in [6] differs from the def-inition we use in this work (which agrees with the one that can be found in theliterature [8, 20, 21, 24]). The “classical” definition of Fourier dimension agreeswith the compact Fourier dimension dim FC of [6, Sec. 1.3] (this can be showed,e.g., using [6, Lem. 1]). The three notions agree if we restrict our attention to thedimension of closed sets. In general, requiring that the measure µ witnessing thatdim F ( A ) > s gives full measure to A is strictly weaker than requiring that µ issupported on A .The fact that dim F = dim FC implies that the Fourier dimension is inner regularfor compact sets, i.e.dim F ( A ) = sup { dim F ( K ) : K ⊂ A and K is compact } . On the other hand, the Fourier dimension is not finitely stable in general: theBernstein set B ⊂ R (see [16, Example 8.24]) is s.t. every closed subset of B or R \ B is countable, and therefore dim F ( B ) = dim F ( R \ B ) = 0. On the other handdim F ( B ∪ R \ B ) = dim F ( R ) = 1 (see also [6, Sec. 1.3]).We can recover countable stability if we restrict our attention to closed sets: Theorem 2.1 ([6, Prop. 5]) . If { A k } k is a finite or countable family of closedsubsets of R d then dim F (cid:16)[ A k (cid:17) = sup k dim F ( A k ) . It is also known that the Fourier dimension does not behave well under H¨oldercontinuous maps: there is a H¨older continuous transformation that maps the Can-tor middle-third set to the interval [0 , Fact 2.2.
The Fourier dimension is invariant under affine invertible transforma-tions.As a consequence of Frostman’s lemma, for every Borel subset A of R d , dim F ( A ) ≤ dim H ( A ) (see [20, Chap. 12]). If dim F ( A ) = dim H ( A ) then A is called Salem set .We denote the collection of Salem subsets of X ⊂ R d with S ( X ).In this work we study the descriptive set-theoretic properties of the family ofclosed Salem subsets of the Euclidean space. For an extended presentation ofdescriptive set theory the reader is referred to [16].Let X be a metric space. It is known that the family of Borel subsets of X can bestratified in a hierarchy, called the Borel hierarchy . Let ω be the first uncountableordinal. The levels of the Borel hierarchy are defined by transfinite recursion on In [6, Ex. 7], the authors show that there is a set X s.t. X is a countable union of compactsets and dim FC ( X ) = dim F ( X ) = 0. However, admitting measures giving full measure to the setwould give X full dimension. N THE DESCRIPTIVE COMPLEXITY OF SALEM SETS 5 ≤ ξ < ω as follows: we start from the families Σ ( X ) and Π ( X ) of the openand the closed subsets of X respectively. Then, for every ξ > Σ ξ ( X ) := (cid:8)S n A n : A n ∈ Π ξ n ( X ) , ξ n < ξ, n ∈ N (cid:9) , Π ξ ( X ) := { X \ A : A ∈ Σ ξ ( X ) } .Moreover, for every ξ , we define ∆ ξ ( X ) := Σ ξ ( X ) ∩ Π ξ ( X ). In particular ∆ ( X ) isthe family of clopen subsets of X . The families Σ ( X ) and Π ( X ) are often writtenresp. F σ ( X ) and G δ ( X ). It is known that B ( X ) = S ξ Σ ξ = S ξ Π ξ = S ξ ∆ ξ ,where B ( X ) denotes the family of Borel subsets of X .Let X and Y be topological spaces and A ⊂ X , B ⊂ Y . We say that A is Wadgereducible to B , and write A ≤ W B , if there is a continuous function f : X → Y s.t. x ∈ A iff f ( x ) ∈ B . It is easy to see that if Γ is among Σ ξ , Π ξ , ∆ ξ then Γ is closedunder continuous preimages, i.e. A ≤ W B and B ∈ Γ ( Y ) implies A ∈ Γ ( X ).Fix a class Γ as above. Assume Y is a Polish (i.e. separable and completelymetrizable) space and B ⊂ Y . We say that B is Γ -hard if A ≤ W B for every A ∈ Γ ( X ), where X is Polish and zero-dimensional (i.e. Hausdorff and with a basisconsisting of clopen sets). If B is Γ -hard and B ∈ Γ ( Y ) then we say that B is Γ -complete .A common technique to show that a set B ⊂ X is Γ -hard is to show that there isa Wadge reduction A ≤ W B , for some A which is already known to be Γ -complete.Standard examples of Γ -complete sets are the following (see [16, Sec. 23.A, p. 179]): Q := { x ∈ N : ( ∀ ∞ m )( x ( m ) = 0) } Σ -complete, N := { x ∈ N : ( ∃ ∞ m )( x ( m ) = 0) } Π -complete, S := { x ∈ N × N : ( ∃ k )( ∃ ∞ m )( x ( k, m ) = 0) } Σ -complete, P := { x ∈ N × N : ( ∀ k )( ∀ ∞ m )( x ( k, m ) = 0) } Π -complete,where ( ∃ ∞ m ) and ( ∀ ∞ m ) mean respectively ( ∀ n )( ∃ m ≥ n ) and ( ∃ n )( ∀ m ≥ n ).For a topological space X , we denote by F ( X ) and K ( X ) respectively the hy-perspaces of closed and compact subsets of X .There is no canonical choice for the topology on F ( X ), and several alternativeshave been explored in the literature [2, 18]. Let U be the collection of sets of theform { F ∈ F ( X ) : F ∩ C = ∅} , where C ranges over all closed subsets of X . The topology having U as a prebaseis called upper topology or upper Vietoris topology ([18, Def. 1.3.1]). In the samespirit, we can define L as the family of sets of the form { F ∈ F ( X ) : F ∩ U = ∅} , where U ranges over the open subsets of X . The topology having U as a prebaseis called lower topology or lower Vietoris topology ([18, Def. 1.3.2]). The Vietoristopology is the topology having as a prebase the family L ∪ U .The Vietoris topology is not always the preferred choice. As an alternative, wecan consider the collection U K of sets of the form { F ∈ F ( X ) : F ∩ K = ∅} , where K ranges over all compact subsets of X . The family U K is a prebase for atopology on F ( X ) called upper Fell topology . We can define the Fell topology on ALBERTO MARCONE AND MANLIO VALENTI F ( X ) as the topology having as a prebase the set U K ∪ L . For this reason, thelower Vietoris topology is often called lower Fell topology . In the following, the Felltopology will be our default choice. For the sake of readability, we will write F U ( X )(resp. F L ( X ), V ( X )) for the hyperspace of closed subsets of X endowed with theupper Fell topology (resp. lower Fell topology, Vietoris topology). Unless otherwisementioned, F ( X ) will be endowed with the Fell topology.Unlike the hyperspace F ( X ), there is a canonical choice for the topology for thehyperspace K ( X ) of compact subsets of X . In fact K ( X ) is usually endowed withthe topology induced from the Vietoris topology on F ( X ).If X is a bounded metric space with distance d , we can define the Hausdorffmetric d H on K ( X ) as follows:d H ( K, L ) := K = L = ∅ diam( X ) if exactly one between K and L is ∅ max { δ ( K, L ) , δ ( L, K ) } otherwisewhere δ ( K, L ) := max x ∈ K d ( x, L ). It is known that the Hausdorff metric d H iscompatible with the Vietoris topology on K ( X ) ([16, Ex. 4.21, p. 25]) and that if X is Polish then so is K ( X ) ([16, Thm. 4.22, p. 25]).The choice of the Vietoris topology is, of course, not the only possible: anytopology on F ( X ) induces a topology on K ( X ). For the sake of readability, we willwrite K F ( X ) (resp. K U ( X ), K L ( X )) for the hyperspace of compact subsets of X ,endowed with the Fell (resp. upper Fell, lower Fell) topology.One of the main reasons why the Vietoris topology is not the canonical choicefor F ( X ) is that it is not paracompact, and hence metrizable , if X is not compact([17, Thm. 2]). On the other hand, if X Polish and locally compact then the Felltopology on F ( X ) gives rise to a Polish compact space and its Borel space is exactlythe Effros-Borel space. The Fell and the Vietoris topologies coincide if X is compact([16, Ex. 12.7, p. 75]).An important topological space is the space of Borel probability measures. If X is a separable metrizable space, we consider the space P ( X ) of Borel probabilitymeasures on X , endowed with the topology generated by the maps µ R f dµ ,with f ∈ C b ( X ) (i.e. f : X → R is continuous and bounded, see [16, Sec. 17.E, p.109]). A basis for the topology on P ( X ) is the family of sets of the form U µ,ε,f ,...,f n := (cid:26) ν ∈ P ( X ) : ( ∀ i ≤ n ) (cid:18)(cid:12)(cid:12)(cid:12)(cid:12)Z X f i dν − Z X f i dµ (cid:12)(cid:12)(cid:12)(cid:12) < ε (cid:19)(cid:27) , where µ ∈ P ( X ), ε >
0, and f i ∈ C b ( X ) for every i . If X is compact metrizablethen so is P ( X ) ([16, Thm. 17.22]). Moreover, if X is Polish then so is P ( X ) ([16,Thm. 17.23]).An important tool in descriptive set theory is Baire category. A set A ⊂ X iscalled nowhere dense if its closure has empty interior, meager if it is the countableunion of nowhere dense sets and comeager if its complement is meager. By theBaire category theorem (see [16, Thm. 8.4]), in every Polish space the intersectionof countably many open dense sets is dense ([16, Prop. 8.1]). In particular everycomeager set is dense (it follows from the definition that a set is comeager iff itcontains a dense G δ set). Intuitively, the max in the definition of δ ( K, L ) is not guaranteed to exist, and two closed setscan be infinitely distant.
N THE DESCRIPTIVE COMPLEXITY OF SALEM SETS 7
We conclude this section with the following lemma:
Lemma 2.3 ([1, Lem. 1.3]) . Let X be Polish and Y metrizable and K σ (i.e. count-able union of compact sets). If F ⊂ X × Y is Σ then proj X ( F ) is also Σ . The complexity of closed Salem subsets of [0 , , H ( A ) > p , dim H ( A ) ≥ p , dim F ( A ) > p and dim F ( A ) ≥ p . Since the upper Felltopology is coarser than the Vietoris topology, obtaining an upper bound for theabove conditions when the hyperspace of compact subsets of [0 ,
1] is endowed withthe upper Fell topology immediately yields an upper bound for the same conditionswhen the hyperspace is endowed with the Vietoris topology instead.
Lemma 3.1.
Let X be a compact subset of R d . The set { ( µ, K, x ) ∈ P ( X ) × K U ( X ) × R : µ ( K ) ≥ x } is closed.Proof. We prove that the complement is open. Let ( µ, K, x ) be s.t. µ ( K ) = x − ε 1] s.t. f ( U ) = 1and f ( R \ V ) = 0.Define the following two sets: U µ,ε/ ,f = (cid:26) ν ∈ P ( X ) : (cid:12)(cid:12)(cid:12)(cid:12)Z f dν − Z f dµ (cid:12)(cid:12)(cid:12)(cid:12) < ε (cid:27) ; U := { H ∈ K U ( X ) : H ⊂ U } . Both U µ,ε/ ,f and U are basic open sets (in the respective topologies). The factthat U is open follows from the fact that X is compact, and therefore so is X \ U .We claim that, for every ( ν, H, y ) ∈ U µ,ε/ ,f × U × B ( x, ε/ 4) we have ν ( H ) < y .Indeed ν ( H ) ≤ ν ( U ) ≤ Z V f dν ≤ Z V f dµ + ε ≤ µ ( V ) + ε < x − ε < y. (cid:3) Notice that the same set is not closed if we consider the lower Fell topology on K ( X ), essentially because X belongs to every non-empty open set U of K L ( X ). Proposition 3.2. • { ( A, p ) ∈ K U ([0 , × [0 , 1] : dim H ( A ) > p } is Σ ; • { ( A, p ) ∈ K U ([0 , × [0 , 1] : dim H ( A ) ≥ p } is Π .Proof. As noticed in the previous section, for Borel (in particular closed) A ⊂ [0 , H ( A ) coincides with the capacitary dimension dim c ( A ).For ease of readability define D ( A ) := { s ∈ [0 , 1] : ( ∃ µ ∈ P ( A ))( ∃ c > ∀ x ∈ R )( ∀ r > µ ( B ( x, r )) ≤ cr s ) } . ALBERTO MARCONE AND MANLIO VALENTI Notice that D ( A ) is downward closed. Recall that dim c ( A ) = sup D ( A ). Clearly µ ( B ( x, r )) ≤ cr s ⇐⇒ µ ([0 , \ B ( x, r )) ≥ − cr s . Observe that the map ( x, r ) [0 , \ B ( x, r ) is continuous when the codomain isendowed with the Vietoris topology. In particular, it is continuous as a function R → K U ([0 , µ ( B ( x, r )) ≤ cr s is closed, hencethe set C := { ( s, c, µ ) : ( ∀ x ∈ R )( ∀ r > µ ( B ( x, r )) ≤ cr s ) } is a closed subset of the product space [0 , × [0 , + ∞ ) × P ( A ). Notice also that µ ∈ P ( A ) ⇐⇒ µ ∈ P ([0 , µ ( A ) ≥ . Since the condition µ ( A ) ≥ A of [0 , Q := { ( s, µ ) ∈ [0 , × P ([0 , ∃ c > µ ∈ P ( A ) ∧ ( s, c, µ ) ∈ C ) } is Σ .Recall that the space P ([0 , D ( A ) = proj [0 , Q is Σ . To conclude the proof we noticethat the conditionsdim c ( A ) > p ⇐⇒ ( ∃ s ∈ Q )( s > p ∧ s ∈ D ( A )) , dim c ( A ) ≥ p ⇐⇒ ( ∀ s ∈ Q )( s < p → s ∈ D ( A ))are Σ and Π respectively. (cid:3) Proposition 3.3. • { ( A, p ) ∈ K U ([0 , × [0 , 1] : dim F ( A ) > p } is Σ ; • { ( A, p ) ∈ K U ([0 , × [0 , 1] : dim F ( A ) ≥ p } is Π .Proof. For the sake of readability, let D ( A ) := { s ∈ [0 , 1] : ( ∃ µ ∈ P ( A ))( ∃ c > ∀ x ∈ R )( | b µ ( x ) | ≤ c | x | − s/ ) } . First of all we notice that the condition | b µ ( x ) | > c | x | − s/ is Σ . To see thisit is enough to show that the map F : P ([0 , × R → R s.t. F ( µ, x ) = | b µ ( x ) | is continuous. Indeed, if that is the case, then the tuple ( µ, x, s, c ) satisfies thecondition | b µ ( x ) | > c | x | − s/ iff it belongs to the preimage of (0 , + ∞ ) via the map( µ, x, s, c ) F ( µ, x ) − c | x | − s/ , which is clearly continuous.Recall that, for each finite Borel measure µ , the Fourier transform b µ is a boundeduniformly continuous function.Notice that the set V µ,ε,x := { ν ∈ P ([0 , | b µ ( x ) − b ν ( x ) | < ε } is open in the topology of P ([0 , ν ∈ V µ,ε,x and let δ s.t. | b µ ( x ) − b ν ( x ) | + δ < ǫ . We claim that the basic open set U ν, δ , cos( x · ) , sin( x · ) is included in V µ,ε,x . In fact, for each η ∈ U ν, δ , cos( x · ) , sin( x · ) we have | b ν ( x ) − b η ( x ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z e − i xt dν ( t ) − Z e − i xt dη ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ≤≤ (cid:12)(cid:12)(cid:12)(cid:12)Z cos( xt ) dν ( t ) − Z cos( xt ) dη ( t ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12)Z sin( xt ) dν ( t ) − Z sin( xt ) dη ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ δ N THE DESCRIPTIVE COMPLEXITY OF SALEM SETS 9 and therefore | b µ ( x ) − b η ( x ) | ≤ | b µ ( x ) − b ν ( x ) | + | b ν ( x ) − b η ( x ) | ≤ | b µ ( x ) − b ν ( x ) | + δ < ε. To conclude the proof of the continuity we show that for each ε > µ and x we can choose δ sufficiently small s.t. for every ( ν, y ) ∈ V µ,δ,x × B ( x, δ ) wehave | b µ ( x ) − b ν ( y ) | < ε . Indeed, by the triangle inequality | b µ ( x ) − b ν ( y ) | ≤ | b µ ( x ) − b ν ( x ) | + | b ν ( x ) − b ν ( y ) | . The first term is bounded by δ by definition of V µ,δ,x . Moreover | b ν ( x ) − b ν ( y ) | = (cid:12)(cid:12)(cid:12)(cid:12)Z e − ixt − e − iyt dν ( t ) (cid:12)(cid:12)(cid:12)(cid:12) ≤≤ Z | cos( xt ) − cos( yt ) | dν ( t ) + Z | sin( xt ) − sin( yt ) | dν ( t ) . By the sum-to-product formulas Z | cos( xt ) − cos( yt ) | dν ( t ) = Z (cid:12)(cid:12)(cid:12)(cid:12) sin (cid:18) ( x + y ) t (cid:19) sin (cid:18) ( x − y ) t (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) dν ( t ) ≤≤ (cid:18) x − y (cid:19) and similarly Z | sin( xt ) − sin( yt ) | dν ( t ) ≤ (cid:18) x − y (cid:19) . hence the claim follows.Since µ ∈ P ( A ) is a closed condition (see the proof of Proposition 3.2), the set { ( s, c, µ ) ∈ [0 , × [0 , + ∞ ) × P ([0 , µ ∈ P ( A ) ∧ | b µ ( x ) | ≤ c | x | − s/ } is closed and, therefore, the set Q := { ( s, µ ) ∈ [0 , × P ([0 , ∃ c > µ ∈ P ( A ) ∧ | b µ ( x ) | ≤ c | x | − s/ ) } is Σ . As in the proof of Proposition 3.2, we can use Lemma 2.3 to conclude thatthe set D ( A ) = proj [0 , Q is Σ and hence the conditionsdim F ( A ) > p ⇐⇒ ( ∃ s ∈ Q )( s > p ∧ s ∈ D ( A )) , dim F ( A ) ≥ p ⇐⇒ ( ∀ s ∈ Q )( s < p → s ∈ D ( A ))are Σ and Π respectively. (cid:3) Theorem 3.4. The set { A ∈ K U ([0 , A ∈ S ([0 , } is Π .Proof. To prove that A ∈ S ([0 , Π condition recall that, for Borel subsets of R d , dim F ( A ) ≤ dim H ( A ). For a closed subset A of [0 , H ( A ) =dim F ( A ) can be written as( ∀ r ∈ Q )(dim H ( A ) > r → dim F ( A ) > r ) . The claim follows from Proposition 3.2 and Proposition 3.3, as both dim H ( A ) > r and dim F ( A ) > r are Σ conditions. (cid:3) We now show that the above conditions are complete for their respective classes(i.e. the upper bounds are tight) when the hyperspace of compact subsets of [0 , , 1] is endowed with the upper Fell topology.The proof of the following Lemma 3.6 exploits the properties of the set E ( α ) of α -well approximable numbers. Definition 3.5 ([8, Sec. 10.3, p. 172]) . For every α ≥ 0, we say that x ∈ [0 , 1] is α -well approximable if there are infinitely many n ∈ N s.t.min m ∈ Z | nx − m | ≤ n − − α . The set of α -well approximable numbers is denoted by E ( α ).As mentioned in the introduction, E ( α ) is a Salem set of dimension 2 / (2 + α ).Notice that, by definition, the set E ( α ) is Π , as it can be written in the form E ( α ) = \ k ∈ N [ n ≥ k G n , where G n := { x ∈ [0 , 1] : min m ∈ Z | nx − m | ≤ n − − α } is a closed set (it is a finiteunion of non-degenerate closed intervals).If α = 0 then, by Dirichlet’s theorem ([8, Ex. 10.8]), E ( α ) = [0 , α > E ( α ) is not closed (because E ( α ) is dense in [0 , 1] but does not havefull dimension).In the construction presented in [4], the author explicitly writes the support S ( α ) of a measure witnessing that dim F ( E ( α )) ≥ / (2 + α ). This, in particular,implies that S ( α ) itself is Salem with dimension 2 / (2 + α ). The set S ( α ) can bewritten as S ( α ) = \ k ∈ N [ k ′ ≤ n ≤ k ′′ G n . In other words, it is obtained from E ( α ) by making the inner union finite, where k ′ and k ′′ depend on k and are strictly increasing. Clearly S ( α ) is closed (as it isthe infinite intersection of closed sets). We can rewrite S ( α ) as follows: S ( α ) = \ k ∈ N S ( k ) ( α )where S ( k ) ( α ) = [ i ≤ M k I i ( α, k )and, for each k , the I i ( α, k ) are disjoint non-degenerate closed intervals.We modify S ( α ) to obtain R ( α ) = \ k ∈ N R ( k ) ( α ) = \ k ∈ N [ j ≤ N k J j ( α, k ) , where each J j ( α, k ) is a non-degenerate closed interval, with the property that R ( k +1) ( α ) ⊂ R ( k ) ( α ), and, moreover, for every i ≤ N k there exists j ≤ N k +13 In [4] it is denoted with S α . N THE DESCRIPTIVE COMPLEXITY OF SALEM SETS 11 s.t. J j ( α, k + 1) ⊂ J i ( α, k ). To this end, define R ( k ) ( α ) inductively as follows: R (0) ( α ) := S (0) ( α ). At stage k + 1, let˜ R ( k +1) ( α ) := S ( k +1) ( α ) ∪ [ n ∈ U k G n , where U k ⊂ N is a finite set of indexes s.t. for every interval j ≤ N k ,Int( J j ( α, k )) ∩ ˜ R ( k +1) ( α ) = ∅ , where Int( · ) denotes the interior. Such a choice of U k is always possible by thedensity of E ( α ). We obtain R ( k +1) ( α ) by considering the finitely many intervalswhose union is ˜ R ( k +1) ( α ) ∩ R ( k ) ( α ) and removing the degenerate ones.Notice that, for every k , S ( k ) ( α ) \ R ( k ) ( α ) is finite. This implies that S ( α ) \ R ( α )is countable and therefore, by Theorem 2.1, dim F ( S ( α )) = dim F ( R ( α )). Notice,moreover, that R ( α ) ⊂ E ( α ), and therefore R is still a Salem set and dim( R ( α )) =2 / (2 + α ). Lemma 3.6. For every p ∈ [0 , there exists a continuous map f p : 2 N → K ([0 , s.t. for every x , f p ( x ) is Salem and dim( f p ( x )) = ( p if x ∈ Q if x / ∈ Q Proof. Recall that Q = { x ∈ N : ( ∀ ∞ k )( x ( k ) = 0) } is Σ -complete.The case p = 0 is trivial (just take the constant map x 7→ ∅ ), so assume p > α ≥ / (2 + α ) = p and consider the Salem set S ( α ) as defined above.For each x ∈ N we define a sequence ( F ( k ) x ) k ∈ N of nested closed sets s.t. each F ( k ) x is a finite union of closed intervals. The idea is to follow the constructionof R ( α ) until we find a k s.t. x ( k ) = 1. If this never happens then in the limitwe obtain R ( α ), which is a Salem set of Fourier dimension p . On the other hand,each time we find a k s.t. x ( k ) = 1 we modify the next step of the construction byreplacing each of the (finitely many) intervals J , . . . , J N k whose union is the k -thlevel of the construction with sufficiently small subintervals H , . . . , H N k , and wereset the construction, starting again a (proportionally scaled down) constructionof R ( α ) on each subinterval H i . By carefully choosing the length of the subintervals H i we can ensure that, if there are infinitely many k s.t. x ( k ) = 1 then F x has nullHausdorff (and hence Fourier) dimension.Formally, if I = [ a, b ] is an interval then we define R ( α, I ) as the fractal obtainedby scaling R ( α ) to the interval I . Notice that, by Fact 2.2, R ( α, I ) is still a Salemset of dimension p .We define F ( k ) x recursively as Stage k = 0 : : F (0) x := [0 , Stage k + 1 : : Let J , . . . , J N k be the disjoint closed intervals s.t. F ( k ) x = S i ≤ N k J i . If x ( k + 1) = 1 then choose, for each i ≤ N k , a (non-degenerate)subinterval H i = [ a i , b i ] ⊂ J i so that X i ≤ N k diam( H i ) − k ≤ − k . Define then F ( k +1) x := S i ≤ N k H i . If x ( k + 1) = 0 then let s ≤ k be largest s.t. x ( s ) = 1 (or s = 0if there is none) and let I , . . . , I N s be the intervals of F ( s ) x . For each i ≤ N s , apply the ( k + 1 − s )-th step of the construction of R ( α, I i ). Define F ( k +1) x := S i ≤ N s R ( k +1 − s ) ( α, I i ).We define the map f p as f p ( x ) := F x = T k ∈ N F ( k ) x . Clearly F x is closed, asintersection of closed sets. To show that f p is continuous, recall that the Vietoristopology is compatible with the Hausdorff metric d H . Fix x ∈ N . For each ε > k large enough so that all the intervals J , . . . , J N k of F ( k ) x havelength ≤ ε . By construction, for every y ∈ N that extends x [ k ] we have F y ∩ J i = ∅ (i.e. none of the intervals is ever removed completely) and F y ⊂ J ∪ . . . ∪ J N k (i.e.nothing is ever added outside of F ( k ) x ). This implies thatd H ( F x , F y ) ≤ max { diam( J i ) : i ≤ N k } ≤ ε, which proves the continuity.If x ∈ Q then x is eventually null (i.e. there are finitely many 1s in x ).Letting s be the largest index s.t. x ( s ) = 1 (or s = 0 if there is none) then F x = S i ≤ N s R ( α, J i ). Each set R ( α, J i ) is a Salem set of dimension p (as we fixed α accordingly). Since the intervals J i are closed and disjoint, using Theorem 2.1,we can conclude that F x is a Salem set of dimension p .On the other hand, if x / ∈ Q then we want to show that dim H ( F x ) = 0. Wewill show that for each s > ε > A n ) n ∈ N of F x s.t. P n ∈ N diam( A n ) s ≤ ε , i.e. for each s > H s ( F x ) = 0.For fixed s and ε we can pick k large enough s.t. 2 − k ≤ s , 2 − k ≤ ε and x ( k + 1) =1. Notice that the intervals ( H i ) i ≤ N k (as defined in the construction of F x ) form acover of F x s.t. X i ≤ N k diam( H i ) s ≤ X i ≤ N k diam( H i ) − k ≤ − k ≤ ε, as desired. (cid:3) Proposition 3.7. For every p < the sets { A ∈ K ([0 , H ( A ) > p } , { A ∈ K ([0 , F ( A ) > p } are Σ -complete.Proof. The hardness is a straightforward corollary of Lemma 3.6: fix q s.t. p < q < Σ -complete subset Q of 2 N . We can consider the map f q : 2 N → S c ([0 , f q ( x )) > p ⇐⇒ x ∈ Q . The completeness follows from Proposition 3.2 and Proposition 3.3. (cid:3) Theorem 3.8. For every p ∈ (0 , , the sets X := { A ∈ K ([0 , H ( A ) ≥ p } ,X := { A ∈ K ([0 , F ( A ) ≥ p } ,X := { A ∈ K ([0 , A ∈ S ([0 , } are Π -complete. N THE DESCRIPTIVE COMPLEXITY OF SALEM SETS 13 Proof. It suffices to prove that the sets X , X and X are Π -hard (the complete-ness follows from Proposition 3.2, Proposition 3.3 and Theorem 3.4 respectively).Recall that P is the Π -complete subset of 2 N × N defined as P := { x ∈ N × N : ( ∀ m )( ∀ ∞ n )( x ( m, n ) = 0) } . Consider the map Φ : 2 N × N → N × N defined asΦ( x )( m, n ) := max i ≤ m x ( i, n ) . It is easy to see that Φ is continuous. Notice also that x ∈ P iff Φ( x ) ∈ P . Onthe other hand, x / ∈ P ⇐⇒ ( ∃ k )( ∀ m ≥ k )( ∃ ∞ n )(Φ( x )( m, n ) = 1) . Intuitively, we are computably modifying the N × N matrix x so that if there is arow of x that contains infinitely many 1s, then, from that row on, every row willcontain infinitely many 1s.To show that X and X are Π -hard, we build a continuous map f : 2 N × N → K ([0 , x ∈ N × N , f ◦ Φ( x ) is a Salem set and dim( f ◦ Φ( x )) ≥ p iff x ∈ P .For every n , let T n := [2 − n − , − n ], q n := p (1 − − n − ) and consider the function f q n : 2 N → S c ([0 , τ n : [0 , → T n and define g n : 2 N → S c ( T n ) as g n := τ n f q n , so that, by Fact 2.2,dim( g n ( y )) = ( q n if y ∈ Q , y / ∈ Q . Let x m be the m -th row of x ∈ N × N . We define f ( x ) := { } ∪ [ m ∈ N g m ( x m ) . Intuitively, we are dividing the interval [0 , 1] into countably many intervals and,on each interval, we are applying the construction we described in the proof ofLemma 3.6 (proportionally scaled down). The continuity of f follows from thecontinuity of each g m . The accumulation point 0 is added to ensure that f ( x ) is aclosed set.Recall that Hausdorff dimension is stable under countable unions, sodim H ( f ( x )) = sup m ∈ N dim H ( g m ( x m )) . Moreover, since the sets { T m } m ∈ N are closed, we can apply Theorem 2.1 andconclude that dim F ( f ( x )) = sup m ∈ N dim F ( g m ( x m )) . Since each g m ( x m ) is Salem we have that f ( x ) is Salem anddim( f ( x )) = sup m ∈ N dim H ( g m ( x m )) = sup m ∈ N dim F ( g m ( x m )) . If x ∈ P then Φ( x ) ∈ P and, for every m , Φ( x ) m ∈ Q . This implies that g m (Φ( x ) m ) is a Salem set of dimension q m and thereforedim( f ◦ Φ( x )) = sup m ∈ N q m = p. On the other hand, if x / ∈ P then there is a k > m ≥ k , Φ( x ) m / ∈ Q and hence dim( g m (Φ( x ) m )) = 0. This implies thatdim( f ◦ Φ( x )) ≤ q k < p. This completes the proof that the sets X and X are Π -hard.With a simple modification of the above argument we can show that X (i.e. thefamily of closed Salem subsets of [0 , Π -hard as well.Let h : 2 N → K ( T ) be a constant map that sends every x ∈ N to a closedsubset of T with Hausdorff dimension p and null Fourier dimension. We define h : 2 N × N → K ([0 , h ( x ) := { } ∪ h ( x ) ∪ [ m ∈ N g m +1 ( x m ) . Notice that, since dim H ( h ( x )) = p , we have dim H ( h ( x )) = p for every x . Inparticular, h ◦ Φ( x ) is Salem iff for every m , dim F ( g m +1 (Φ( x ) m )) = q m +1 , i.e. h ◦ Φ( x ) is a Salem set (with dimension p ) iff x ∈ P . (cid:3) This shows that the upper bounds we obtained in Proposition 3.2, Proposition 3.3and Theorem 3.4 are sharp. In particular, since K ([0 , , 1] is not a Polish space (inthe relative topology). This follows from [16, Thm. 3.11], as a subset of a Polishspace is Polish iff it is G δ .Notice that, if we endow S c ([0 , K U ([0 , { ( A, p ) ∈ S c ([0 , × [0 , 1] : dim( A ) > p } is Σ , { ( A, p ) ∈ S c ([0 , × [0 , 1] : dim( A ) ≥ p } is Π . Moreover, the proofs of Proposition 3.7 and Theorem 3.8 show that, for every p < q > Q ≤ W { A ∈ S c ([0 , A ) > p } ,P ≤ W { A ∈ S c ([0 , A ) ≥ q } . However we cannot say that they are complete for their respective classes, becausethe definition of completeness requires the ambient space to be Polish, and S c ([0 , A is based on an estimate on the decayof the Fourier transform of a probability measure supported on A . In particulardim F ( A ) = sup { dim F ( µ ) : µ ∈ P ( A ) } . This is equivalent to let µ range over finite(non-trivial) Radon measures on A , as the estimate on the decay of the Fouriertransform is only up to a multiplicative constant. One may wonder whether it ispossible to strengthen this condition by defining the Fourier dimension of A assup { s ∈ [0 , 1] : ( ∃ µ ∈ P ( A ))( ∀ x ∈ R )( | b µ ( x ) | ≤ | x | − s/ ) } . The Π -completeness of S c ([0 , P ([0 , , ∞ ) × P ([0 , c in the condition on the decay of the Fouriertransform would imply that S c ([0 , Π (as the projection of a closed set along N THE DESCRIPTIVE COMPLEXITY OF SALEM SETS 15 a compact space is closed, see the proofs of Proposition 3.2 and Proposition 3.3),and therefore not Π -complete.4. The complexity of closed Salem subsets of [0 , d Let us now turn our attention to the family of closed Salem subsets of [0 , d . Proposition 4.1. For every d ≥ : (1) { ( A, p ) ∈ K U ([0 , d ) × [0 , d ] : dim H ( A ) > p } is Σ ; (2) { ( A, p ) ∈ K U ([0 , d ) × [0 , d ] : dim H ( A ) ≥ p } is Π ; (3) { ( A, p ) ∈ K U ([0 , d ) × [0 , d ] : dim F ( A ) > p } is Σ ; (4) { ( A, p ) ∈ K U ([0 , d ) × [0 , d ] : dim F ( A ) ≥ p } is Π ; (5) { A ∈ K U ([0 , d ) : A ∈ S ([0 , d ) } is Π .Proof. For the first two points, the proof is a straightforward adaptation of theproof of Proposition 3.2. Indeed, recall that Frostman’s lemma holds for Borelsubsets of R d ([20, Thm. 8.8]), hence we can characterize the Hausdorff dimensionby means of the capacitary dimension. Moreover, since [0 , d is compact, thecondition µ ( B ( x, r )) ≤ cr s is closed and the space P ([0 , d ) is compact. Thereforedim c ( A ) is the supremum of a Σ set, from which the claim follows.Similarly, points 3 and 4 follow by adapting the proof of Proposition 3.3. Indeedthe map F := ( µ, x ) 7→ | b µ ( x ) | is continuous and the condition | b µ ( x ) | > c | x | − t/ isopen, therefore the Fourier dimension is the supremum of a Σ set.Finally the last point can be proved by following the proof of Theorem 3.4 andusing points 1 and 3. (cid:3) The fact that the lower bounds for the complexity of the above sets are tightdoes not come as a corollary of the results in the 1-dimensional case. Indeed, itis well known that the Fourier dimension is sensitive to the ambient space: any m -dimensional hyperplane has null Fourier dimension when seen as a subset of R d ,with d > m (in particular, the unit interval [0 , 1] has full Fourier dimension if seenas a subset of itself or of R , but it has null Fourier dimension if seen as a subset of R ).We will instead prove a d -dimensional analogue of Lemma 3.6. In recent work,Fraser and Hambrook ([9]) presented a construction of a Salem subset of [0 , d ofdimension p , for every p ∈ [0 , d ]. Definition 4.2 ([9]) . Let K be a number field of degree d , i.e. K is a field extensionof Q and dim Q K = d . Let B = { ω , . . . , ω d − } be an integral basis for K . Wecan identify Q d with K by mapping a vector q = ( q , . . . , q n − ) to P i 0) is Salem of dimension d is not explicitly mentioned in [9],but a simple proof was suggested by Hambrook (personal communication): indeedit is enough to notice that, for every α and every ε > E ( K, B, α + ε ) ⊂ E ( K, B, α ),and therefore the claim follows from the monotonicity of the Fourier dimension.Notice that, in general, the set E ( K, B, α ) is not closed but Π . Analogouslyto the one-dimensional case, the proof of Theorem 4.3 shows that there is a closedSalem subset S ( K, B, α ) of E ( K, B, α ) with dimension 2 d/ (2 + α ). To prove thefollowing Lemma 4.4 we cannot proceed as in the one-dimension case, as we do notknow whether E ( K, B, α ) is dense in [0 , d . Lemma 4.4. Fix d > . For every p ∈ [0 , d ] there exists a continuous map f p : 2 N → K ([0 , d ) s.t. for every x , f p ( x ) is Salem and dim( f p ( x )) = ( p if x ∈ Q if x / ∈ Q Proof. The idea of the proof is similar to the one of Proposition 3.7: given x ∈ N ,we define a closed set F x by following the construction of the set S ( K, B, α ), havingcare of controlling the Hausdorff dimension whenever x ( k ) = 1.Formally, let p > x 7→ ∅ ) and let α s.t. 2 d/ (2 + α ) = p .Fix K and B as in Definition 4.2. For the sake of readability, let S ( α ) := S ( K, B, α ). We can write S ( α ) as intersection of closed nested sets S ( k ) ( α ) definedas S ( k ) ( α ) := { y ∈ [0 , d : d ( y, S ( α )) ≤ − k } . Clearly, S ( k ) ( α ) is closed with non-empty interior.For each non-degenerate hypercube C , define S ( α, C ) := τ ( S ( α )), where τ is asimilarity transformation that maps [0 , d onto C , and S ( k ) ( α, C ) accordingly.We define F ( k ) x recursively, ensuring that, for each k , F ( k ) x is closed and hasnon-empty interior, and F ( k +1) x ⊂ F ( k ) x : Stage k = 0 : F (0) x := C := [0 , d , P := ∅ ; Stage k + 1 : If x ( k + 1) = 1, let P k := { p ( k ) i } i ≤ N k be a finite set of points in F ( k ) x s.t. for each t ∈ F ( k ) x there exists i ≤ N k s.t. | t − p ( k ) i | ≤ − ( k +1) . Let C k be the largest (non-degenerate) hypercube contained in F ( k ) x . Define F ( k +1) x := S (0) ( α, C k ) ∪ P k . If x ( k + 1) = 0 then let s < k be largest s.t. x ( s + 1) = 1 (or s = 0 ifthere is none). We define F ( k +1) x := S ( k +1 − s ) ( α, C s ) ∪ P s . Define f p := x F x = T k ∈ N F ( k ) x . Clearly F x is closed, as intersection of closedsets. The continuity of the map f p is guaranteed by the fact that for each k ,d H ( F ( k ) x , F ( k +1) x ) ≤ − ( k +1) . This follows from our choice of P k in the first case, and d H ( S ( k ) ( α ) , S ( k +1) ( α )) ≤ − ( k +1) in the second case.Adapting the proof of Lemma 3.6, it is possible to show that F x is Salem andthat dim( F x ) = p iff x ∈ Q . (cid:3) N THE DESCRIPTIVE COMPLEXITY OF SALEM SETS 17 From Lemma 4.4 we can derive the following results, as we did with their ana-logues in the previous section. Proposition 4.5. For every p ∈ [0 , d ) the sets { A ∈ K ([0 , d ) : dim H ( A ) > p } , { A ∈ K ([0 , d ) : dim F ( A ) > p } are Σ -complete. Theorem 4.6. For every p ∈ (0 , d ] , the sets { A ∈ K ([0 , d ) : dim H ( A ) ≥ p } , { A ∈ K ([0 , d ) : dim F ( A ) ≥ p } , { A ∈ K ([0 , d ) : A ∈ S ([0 , d ) } are Π -complete. We now discuss an alternative proof for the Π -completeness of the closed Salemsubsets of [0 , d : as noticed in the introduction, using a theorem of Gatesoupe [10]we can show that if A ⊂ [0 , 1] is Salem with dimension α then the set r ( A ) := { x ∈ [ − , d : | x | ∈ A } is a Salem set with dimension d − α . Using Fact 2.2, we canmap r ( A ) to a Salem subset of C , for every d -dimensional cube C . Moreover, foreach p ∈ [ d − , d ] there is a compact set Y p ⊂ [0 , d with null Fourier dimensionand Hausdorff dimension p (e.g. consider the cartesian product of [0 , d − with anon-empty subset of [0 , 1] with Hausdorff dimension p − ( d − { C n } n ∈ N bea family of mutually disjoint closed cubes s.t. • C n ⊂ [0 , d , • S n ∈ N C n = { } ∪ S n ∈ N C n .We mimic the proof of Theorem 3.8 and construct a set X n within each C n , whereeach X n is the image, under a similarity transformation, of either Y p or a radialset of the type r ( A n ) for some A n ⊂ [0 , X := { } ∪ S n ∈ N X n .Since the cubes are disjoint we have that dim F ( X ) = sup n dim F ( X n ) and therefore,by carefully choosing the dimensions of each A n , we obtain the results on thecomplexities.Notice however that each r ( A n ) has Fourier (and hence Hausdorff) dimension atleast d − 1. Therefore, while this argument suffices to show the Π -completenessof { A ∈ K ([0 , d ) : A ∈ S ([0 , d ) } , it gives no information on the complexity ofthe first two sets listed in Theorem 4.6 when p < d − 1. On the other hand, theconstruction presented in Lemma 4.4 has the advantage to work for every p ∈ [0 , d ].5. The complexity of closed Salem subsets of R d Let us now turn our attention to the closed Salem subsets of R d . In this section,we determine the descriptive complexity of the conditions dim H ( A ) > p , dim H ( A ) ≥ p , dim F ( A ) > p , dim F ( A ) ≥ p , A ∈ S ( R d ), when A is a closed subset of R d and p ∈ R .The hardness results lift easily from the compact cases. Proposition 5.1. For every p ∈ (0 , d ] and every q ∈ [0 , d ) , we have • { A ∈ F ( R d ) : dim H ( A ) > q } is Σ -hard; • { A ∈ F ( R d ) : dim H ( A ) ≥ p } is Π -hard; • { A ∈ F ( R d ) : dim F ( A ) > q } is Σ -hard; • { A ∈ F ( R d ) : dim F ( A ) ≥ p } is Π -hard; • { A ∈ F ( R d ) : A ∈ S ( R d ) } is Π -hard.Proof. This is a corollary of Proposition 4.5 and Theorem 4.6. Indeed, since theFourier and Hausdorff dimensions of A ⊂ [0 , d do not change if we see A as asubset of R d , it is enough to notice that the inclusion map K ([0 , d ) ֒ → F ( R d ) iscontinuous. (cid:3) Notice that, since the inclusion K ([0 , d ) ֒ → V ( R d ) is continuous as well, thesame proof provides a lower bound for the above conditions when the hyperspace F ( R d ) is endowed with the Vietoris topology. However, since V ( R d ) is not Polish,we cannot say that the conditions are hard for their respective classes.As in the previous sections, we obtain the upper bounds endowing F ( R d ) withthe upper Fell topology. This will yield, as a corollary, that each of the aboveconditions is complete for its respective class, when F ( R d ) is endowed with the Felltopology (in case of the upper Fell or Vietoris topology we only obtain a Wadge-equivalence).Since the proofs of Proposition 3.2, Proposition 3.3 and Proposition 4.1 exploitthe compactness of the ambient space, some extra care is needed when working ina non-compact environment. Lemma 5.2. • { ( K, p ) ∈ K ( R d ) × [0 , d ] : dim H ( K ) > p } is Σ ; • { ( K, p ) ∈ K ( R d ) × [0 , d ] : dim H ( K ) ≥ p } is Π .Proof. Define D ( K ) := { s ∈ [0 , d ] : ( ∃ µ ∈ P ( K ))( ∃ c > ∀ x ∈ R d )( ∀ r > µ ( B ( x, r )) ≤ cr s ) } and recall that dim c ( K ) = sup D ( K ). For every n , let K n := B ( , n ). Observethat µ ∈ P ( K ) ⇐⇒ µ ∈ P ( R d ) ∧ µ ( K ) ≥ ∧ ( ∃ n ∈ N )( µ ( K n ) ≥ . We can therefore rewrite D ( K ) as follows D ( K ) = { s ∈ [0 , d ] : ( ∃ µ ∈ P ( R d ))( ∃ c > ∃ n ∈ N )( µ ( K ) ≥ ∧ µ ( K n ) ≥ ∧ ( ∀ x ∈ R d )( ∀ r > µ ( B ( x, r )) ≤ cr s )) } . In particular µ ( K n ) ≥ µ ⊂ K n , hence µ ( B ( x, r )) ≤ cr s ⇐⇒ µ ( H ) ≥ − cr s , where H := B ( , n + x + r ) \ B ( x, r ). It is routine to prove that the function ϕ : N × R + × R d → K ( R d ) that sends ( n, r, x ) to the above-defined H is continuous.Notice that if we had set H = K \ B ( x, r ) then the resulting map would not becontinuous. This motivates the use of K n in the above characterization of D ( K ).By [16, Ex. 17.29, p. 114], for every separable metric space X the set { ( µ, K, a ) ∈ P ( X ) × K ( X ) × R : µ ( K ) ≥ a } is closed. In particular the condition µ ( B ( x, r )) ≤ cr s is closed and the set Q := { ( s, µ ) ∈ [0 , d ] × P ( R d ) : ( ∃ c > ∃ n ∈ N )( µ ( K ) ≥ ∧ µ ( K n ) ≥ ∧ ( ∀ x ∈ R d )( ∀ r > µ ( B ( x, r )) ≤ cr s )) } N THE DESCRIPTIVE COMPLEXITY OF SALEM SETS 19 is Σ .Notice that we can equivalently consider Q as a subset of [0 , d ] × S n ∈ N P ( K n ).In particular, D ( K ) is the projection of a Σ set along a metrizable and K σ space(as P ( X ) is compact if X is). Therefore, using Lemma 2.3 we can conclude that D ( K ) is Σ and that the conditionsdim c ( K ) > p ⇐⇒ ( ∃ s ∈ Q )( s > p ∧ s ∈ D ( K )) , dim c ( K ) ≥ p ⇐⇒ ( ∀ s ∈ Q )( s < p → s ∈ D ( K ))are Σ and Π respectively. (cid:3) Lemma 5.3. The set { ( A, B ) ∈ F ( R d ) × F ( R d ) : B ⊂ A } is Π .Proof. It suffices to show that the complement of the set is open. If B A , fix x ∈ B \ A and let ε := d ( x, A ) > 0. Let U := { F ∈ F ( R d ) : F ∩ B ( x, ε/ 2) = ∅} and U := { F ∈ F ( R d ) : F ∩ B ( x, ε/ = ∅ } . Then ( A, B ) ∈ U × U and every( A ′ , B ′ ) ∈ U × U is s.t. B ′ A ′ . (cid:3) Theorem 5.4. The sets X := { ( A, p ) ∈ F U ( R d ) × [0 , d ] : dim H ( A ) > p } ,X := { ( A, p ) ∈ F U ( R d ) × [0 , d ] : dim H ( A ) ≥ p } are Σ and Π respectively. In particular, for every p ∈ [0 , d ) and q ∈ (0 , d ] , thesets { A ∈ F ( R d ) : dim H ( A ) > p } , { A ∈ F ( R d ) : dim H ( A ) ≥ q } are Σ -complete and Π -complete respectively.Proof. Notice that, as a consequence of the countable stability of the Hausdorffdimension, we havedim H ( A ) = sup { dim H ( K ) : K ⊂ A and K is compact } , and thereforedim H ( A ) > p ⇐⇒ ( ∃ K ∈ F ( R d ))( K ⊂ A ∧ K ∈ K ( R d ) ∧ dim H ( K ) > p ) . By Lemma 5.3 and Lemma 5.2 we have that • K ⊂ A is Π ; • K ∈ K ( R d ) is Σ , because it is equivalent to ( ∃ n )( K ⊂ B ( , n )); • dim H ( K ) > p is Σ .This shows that the set X is the projection of a Σ set along F ( R d ). Since F ( R d )is compact, we can use Lemma 2.3 and conclude that X is Σ .Moreover, since dim H ( A ) ≥ p iff ( ∀ r ∈ Q )( r < p → dim H ( A ) > r ), this alsoshows that X is Π . The completeness follows from Proposition 5.1. (cid:3) With a similar strategy, we can characterize the upper bounds for the Fourierdimension: Theorem 5.5. The sets X := { ( A, p ) ∈ F U ( R d ) × [0 , d ] : dim F ( A ) > p } ,X := { ( A, p ) ∈ F U ( R d ) × [0 , d ] : dim F ( A ) ≥ p } are Σ and Π respectively. In particular, for every p ∈ [0 , d ) and q ∈ (0 , d ] , thesets { A ∈ F ( R d ) : dim F ( A ) > p } , { A ∈ F ( R d ) : dim F ( A ) ≥ q } are Σ -complete and Π -complete respectively.Proof. Notice that the condition( ∀ x ∈ R d )( | b µ ( x ) | ≤ c | x | − s/ )is closed, as the map ( µ, x, s, c ) 7→ | b µ ( x ) | − c | x | − s/ is continuous (see also the proofof Proposition 3.3). In particular this implies that the sets { ( K, p ) ∈ K ( R d ) × [0 , d ] : dim F ( K ) > p } , { ( K, p ) ∈ K ( R d ) × [0 , d ] : dim F ( K ) ≥ p } are Σ and Π respectively.Since the Fourier dimension is inner regular for compact sets, we can writedim F ( A ) > p ⇐⇒ ( ∃ K ∈ F ( R d ))( K ⊂ A ∧ K ∈ K ( R d ) ∧ dim F ( K ) > p ) . Using Lemma 5.3 and the fact that K ( R d ) is a Σ subset of F ( R d ) we havethat X is the projection of a Σ set along F ( R d ). This implies that X is Σ and X is Π (as in the proof of Theorem 5.4). The completeness follows fromProposition 5.1. (cid:3) Theorem 5.6. The set { A ∈ F ( R d ) : A ∈ S ( R d ) } is Π -complete.Proof. Using Theorem 5.4 and Theorem 5.5 we have that, for every p , the condi-tions dim H ( A ) > p and dim F ( A ) > p are Σ . The fact that { A ∈ F ( R d ) : A ∈ S ( R d ) } is Π follows as in the proof of Theorem 3.4, while the completeness followsfrom Proposition 5.1. (cid:3) Final remarks Let X be [0 , d or R d , for some d ≥ 1. Notice that the set S c ( X ) is comea-ger in V ( X ). Indeed, the set { K ∈ V ( X ) : dim H ( K ) ≤ } ⊂ S c ( X ) is Π by Proposition 3.2 (and its higher-dimensional analogues), and dense because itcontains the set { K ∈ V ( X ) : K is finite } , which is dense. The same argu-ment also shows that for every p the sets { K ∈ V ( X ) : dim H ( K ) ≤ p } and { K ∈ V ( X ) : dim F ( K ) ≤ p } are comeager.Recall that if Γ is a level in the Borel hierarchy, we say that f : X → Y is Γ -measurable if, for every open U ⊂ Y , f − ( U ) ∈ Γ ( X ). Proposition 4.5,Theorem 5.4 and Theorem 5.5 show that the maps dim H : F ( X ) → R and dim F : F ( X ) → R are Σ -measurable. Using [16, Thm. 24.3, p. 190], this is equivalent to both dim H and dim F being Baire class 2. 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Department of Mathematics, Computer Science and PhysicsUniversity of UdineUdine, UD 33100, IT Email address : [email protected] Department of Mathematics, Computer Science and PhysicsUniversity of UdineUdine, UD 33100, IT Email address ::