On the descriptive complexity of Salem sets Manlio Valenti manlio . - - PowerPoint PPT Presentation

On the descriptive complexity of Salem sets

Department of Mathematics, Computer Science, Physics University of Udine Joint work with Alberto Marcone

CCA Sep, 11, 2020

Manlio Valenti manlio.valenti@uniud.it

Question

During the IMS Graduate Summer School in Logic in 2018, Slaman asked:

“What is the descriptive complexity of the family

• f closed Salem sets in [0, 1]?”

Manlio Valenti On the descriptive complexity of Salem sets 1/20

Hausdorfg dimension

Standard notion in geometric measure theory. Using Frostman’s lemma, the Hausdorfg dimension of A

d

can be written as A s A c x

d

r B x r crs In other words, it coincides with the capacitary dimension.

Manlio Valenti On the descriptive complexity of Salem sets 2/20

Hausdorfg dimension

Standard notion in geometric measure theory. Using Frostman’s lemma, the Hausdorfg dimension of A ∈ B(Rd) can be written as dimH(A) = sup{s : (∃µ ∈ P(A)) (∃c > 0) (∀x ∈ Rd) (∀r > 0) (µ(B(x, r)) ≤ crs)} In other words, it coincides with the capacitary dimension.

Manlio Valenti On the descriptive complexity of Salem sets 2/20

Fourier dimension

Let be a fjnite Borel measure. Fourier transform of :

d

defjned as e

i x d

x The Fourier dimension of A

d is defjned as F A

s 0 d A c x

d

x c x

s 2

Manlio Valenti On the descriptive complexity of Salem sets 3/20

Fourier dimension

Let µ be a fjnite Borel measure. Fourier transform of µ: µ: Rd → C defjned as

• µ(ξ) :=

∫ e−i ξ·x dµ(x) The Fourier dimension of A

d is defjned as F A

s 0 d A c x

d

x c x

s 2

Manlio Valenti On the descriptive complexity of Salem sets 3/20

Fourier dimension

Let µ be a fjnite Borel measure. Fourier transform of µ: µ: Rd → C defjned as

• µ(ξ) :=

∫ e−i ξ·x dµ(x) The Fourier dimension of A ⊂ Rd is defjned as dimF(A) := sup{s ∈ [0, d] : (∃µ ∈ P(A)) (∃c > 0) (∀x ∈ Rd) (| µ(x)| ≤ c|x|−s/2)}

Manlio Valenti On the descriptive complexity of Salem sets 3/20

Salem sets

Proposition (Folklore?)

For every A ∈ B(Rd) we have dimF(A) ≤ dimH(A) This shows that the Fourier dimension can be used to obtain lower bounds for the Hausdorfg dimension. A set A s.t. A

F A is called Salem set.

Manlio Valenti On the descriptive complexity of Salem sets 4/20

Salem sets

Proposition (Folklore?)

For every A ∈ B(Rd) we have dimF(A) ≤ dimH(A) This shows that the Fourier dimension can be used to obtain lower bounds for the Hausdorfg dimension. A set A s.t. dimH(A) = dimF(A) is called Salem set.

Manlio Valenti On the descriptive complexity of Salem sets 4/20

Salem sets

Examples: , 0 1 are Salem subsets of 0 1 . Cantor middle-third set is not Salem: it has 2 3 and

F

Deterministic (non-trivial) Salem sets are rare. Classic example: Jarník’s fractal.

Theorem (Jarník 1928, Besicovitch 1934, Kaufmann 1981)

For every x 0 1 x is

• well approximable

2 2

Manlio Valenti On the descriptive complexity of Salem sets 5/20

Salem sets

Examples: ∅, [0, 1] are Salem subsets of [0, 1]. Cantor middle-third set is not Salem: it has 2 3 and

F

Deterministic (non-trivial) Salem sets are rare. Classic example: Jarník’s fractal.

Theorem (Jarník 1928, Besicovitch 1934, Kaufmann 1981)

For every x 0 1 x is

• well approximable

2 2

Manlio Valenti On the descriptive complexity of Salem sets 5/20

Salem sets

Examples: ∅, [0, 1] are Salem subsets of [0, 1]. Cantor middle-third set is not Salem: it has dimH = log(2) log(3) and dimF = 0 Deterministic (non-trivial) Salem sets are rare. Classic example: Jarník’s fractal.

Theorem (Jarník 1928, Besicovitch 1934, Kaufmann 1981)

For every x 0 1 x is

• well approximable

2 2

Manlio Valenti On the descriptive complexity of Salem sets 5/20

Salem sets

Examples: ∅, [0, 1] are Salem subsets of [0, 1]. Cantor middle-third set is not Salem: it has dimH = log(2) log(3) and dimF = 0 Deterministic (non-trivial) Salem sets are rare. Classic example: Jarník’s fractal.

Theorem (Jarník 1928, Besicovitch 1934, Kaufmann 1981)

For every x 0 1 x is

• well approximable

2 2

Manlio Valenti On the descriptive complexity of Salem sets 5/20

Salem sets

Examples: ∅, [0, 1] are Salem subsets of [0, 1]. Cantor middle-third set is not Salem: it has dimH = log(2) log(3) and dimF = 0 Deterministic (non-trivial) Salem sets are rare. Classic example: Jarník’s fractal.

Theorem (Jarník 1928, Besicovitch 1934, Kaufmann 1981)

For every α ≥ 0 dim({x ∈ [0, 1] : x is α-well approximable}) = 2 2 + α

Manlio Valenti On the descriptive complexity of Salem sets 5/20

Wadge reducibility

We will locate the family of closed Salem sets in the Borel hierarchy Let X and Y be Polish spaces. A X is Wadge reducible to B Y (A

W B) if there is a

continuous map f X Y s.t. x A f x B B is called

• hard if, for every A

2 , A

W B.

B is called

• complete if it is
• hard and B

Y

Manlio Valenti On the descriptive complexity of Salem sets 6/20

Wadge reducibility

We will locate the family of closed Salem sets in the Borel hierarchy Let X and Y be Polish spaces. A ⊂ X is Wadge reducible to B ⊂ Y (A ≤W B) if there is a continuous map f : X → Y s.t. x ∈ A ⇐ ⇒ f (x) ∈ B B is called

• hard if, for every A

2 , A

W B.

B is called

• complete if it is
• hard and B

Y

Manlio Valenti On the descriptive complexity of Salem sets 6/20

Wadge reducibility

We will locate the family of closed Salem sets in the Borel hierarchy Let X and Y be Polish spaces. A ⊂ X is Wadge reducible to B ⊂ Y (A ≤W B) if there is a continuous map f : X → Y s.t. x ∈ A ⇐ ⇒ f (x) ∈ B B is called Γ-hard if, for every A ∈ Γ(2N), A ≤W B. B is called Γ-complete if it is Γ-hard and B ∈ Γ(Y)

Manlio Valenti On the descriptive complexity of Salem sets 6/20

Salem sets in [0, 1]

Work in K([0, 1]): the hyperspace of compact subsets of [0, 1].

Lemma (Marcone, Reimann, Slaman, V.)

A p K 0 1 0 1 A p is

2;

A p K 0 1 0 1 A p is

3;

A p K 0 1 0 1

F A

p is

2;

A p K 0 1 0 1

F A

p is

3.

The proof relies on the compactness of the ambient space 0 1 .

Manlio Valenti On the descriptive complexity of Salem sets 7/20

Salem sets in [0, 1]

Work in K([0, 1]): the hyperspace of compact subsets of [0, 1].

Lemma (Marcone, Reimann, Slaman, V.)

• {(A, p) ∈ K([0, 1]) × [0, 1] : dimH(A) > p} is Σ0

2;

• {(A, p) ∈ K([0, 1]) × [0, 1] : dimH(A) ≥ p} is Π0

3;

• {(A, p) ∈ K([0, 1]) × [0, 1] : dimF(A) > p} is Σ0

2;

• {(A, p) ∈ K([0, 1]) × [0, 1] : dimF(A) ≥ p} is Π0

3.

The proof relies on the compactness of the ambient space [0, 1].

Manlio Valenti On the descriptive complexity of Salem sets 7/20

Salem sets in [0, 1]

Lemma (Marcone, Reimann, Slaman, V.)

For every p ∈ [0, 1] there is a continuous (in fact computable) map fp : 2N → S ([0, 1]) s.t. dim(fp(x)) = { p if x ∈ Q2 if x / ∈ Q2 where Q2 = {x ∈ 2N : (∀∞n)(x(n) = 0)} is Σ0

2-complete.

Manlio Valenti On the descriptive complexity of Salem sets 8/20

Salem sets in [0, 1]

Theorem (Marcone, Reimann, Slaman, V.)

The sets {(A, p) ∈ K([0, 1]) × [0, 1) : dimH(A) > p}, {(A, p) ∈ K([0, 1]) × [0, 1) : dimF(A) > p} are Σ0

2-complete. Moreover the sets

{(A, p) ∈ K([0, 1]) × (0, 1] : dimH(A) ≥ p}, {(A, p) ∈ K([0, 1]) × (0, 1] : dimF(A) ≥ p}, {A ∈ K([0, 1]) : A ∈ S ([0, 1])} are Π0

3-complete.

Manlio Valenti On the descriptive complexity of Salem sets 9/20

How about closed subset of [0, 1]d?

The upper bounds on the complexities are the same (the proof is based on the compactness of the ambient space). Problem: the Fourier dimension is sensitive to the ambient space. If a set A is contained in a m-dimensional hyperplane (with m d) then

F A

0. Some “curvature” is necessary to have positive Fourier dimension.

Manlio Valenti On the descriptive complexity of Salem sets 10/20

How about closed subset of [0, 1]d?

The upper bounds on the complexities are the same (the proof is based on the compactness of the ambient space). Problem: the Fourier dimension is sensitive to the ambient space. If a set A is contained in a m-dimensional hyperplane (with m d) then

F A

0. Some “curvature” is necessary to have positive Fourier dimension.

Manlio Valenti On the descriptive complexity of Salem sets 10/20

How about closed subset of [0, 1]d?

The upper bounds on the complexities are the same (the proof is based on the compactness of the ambient space). Problem: the Fourier dimension is sensitive to the ambient space. If a set A is contained in a m-dimensional hyperplane (with m < d) then dimF(A) = 0. Some “curvature” is necessary to have positive Fourier dimension.

Manlio Valenti On the descriptive complexity of Salem sets 10/20

Solutions?

We can exploit a “higher-dimensional analogue” of Jarník’s fractal, recently defjned by Fraser and Hambrook [4].

Theorem (Fraser, Hambrook)

For every 0, the set E K B is a Salem set of dimension 2d 2 .

Manlio Valenti On the descriptive complexity of Salem sets 11/20

Solutions?

We can exploit a “higher-dimensional analogue” of Jarník’s fractal, recently defjned by Fraser and Hambrook [4].

Theorem (Fraser, Hambrook)

For every α ≥ 0, the set E(K, B, α) is a Salem set of dimension 2d/(2 + α).

Manlio Valenti On the descriptive complexity of Salem sets 11/20

Salem sets in [0, 1]d

Similarly to the one-dimensional case we have:

Lemma (Marcone, V.)

For every p ∈ [0, d] there exists a continuous (in fact computable) map fp : 2N → S ([0, 1]d) s.t. dim(fp(x)) = { p if x ∈ Q2 if x / ∈ Q2 where Q2 = {x ∈ 2N : (∀∞n)(x(n) = 0)} is Σ0

2-complete.

Manlio Valenti On the descriptive complexity of Salem sets 12/20

Salem sets in [0, 1]d

Theorem (Marcone, V.)

For every d ≥ 1, the sets {(A, p) ∈ K([0, 1]d) × [0, d) : dimH(A) > p}, {(A, p) ∈ K([0, 1]d) × [0, d) : dimF(A) > p} are Σ0

2-complete. Moreover the sets

{(A, p) ∈ K([0, 1]d) × (0, d] : dimH(A) ≥ p}, {(A, p) ∈ K([0, 1]d) × (0, d] : dimF(A) ≥ p}, {A ∈ K([0, 1]d) : A ∈ S ([0, 1]d)} are Π0

3-complete.

Manlio Valenti On the descriptive complexity of Salem sets 13/20

Relaxing compactness

Do things change if we move to Rd?

Both Hausdorfg and Fourier dimensions are preserved when moving from 0 1 d to

d.

In particular, every Salem set of 0 1 d is still Salem when seen as a subset of

d.

Hardness results (lower bounds) are corollaries, while upper bounds are more delicate.

Manlio Valenti On the descriptive complexity of Salem sets 14/20

Relaxing compactness

Do things change if we move to Rd?

Both Hausdorfg and Fourier dimensions are preserved when moving from [0, 1]d to Rd. In particular, every Salem set of [0, 1]d is still Salem when seen as a subset of Rd. Hardness results (lower bounds) are corollaries, while upper bounds are more delicate.

Manlio Valenti On the descriptive complexity of Salem sets 14/20

Relaxing compactness

Do things change if we move to Rd?

Both Hausdorfg and Fourier dimensions are preserved when moving from [0, 1]d to Rd. In particular, every Salem set of [0, 1]d is still Salem when seen as a subset of Rd. Hardness results (lower bounds) are corollaries, while upper bounds are more delicate.

Manlio Valenti On the descriptive complexity of Salem sets 14/20

Topology on F(Rd)

There is no “canonical” topology on the hyperspace F

d of closed

subsets of

d.

We considered both the Vietoris topology

V and the Fell topology F.

Vietoris Fell

+ Familiar for topologists + F

d F

is Polish and com- pact + It is “the same” topology we put on K 0 1 d + Generates a standard Borel space

• Not metrizable if the ambient

space is not compact Coarser than Vietoris topology

Manlio Valenti On the descriptive complexity of Salem sets 15/20

Topology on F(Rd)

There is no “canonical” topology on the hyperspace F(Rd) of closed subsets of Rd. We considered both the Vietoris topology

V and the Fell topology F.

Vietoris Fell

+ Familiar for topologists + F

d F

is Polish and com- pact + It is “the same” topology we put on K 0 1 d + Generates a standard Borel space

• Not metrizable if the ambient

space is not compact Coarser than Vietoris topology

Manlio Valenti On the descriptive complexity of Salem sets 15/20

Topology on F(Rd)

There is no “canonical” topology on the hyperspace F(Rd) of closed subsets of Rd. We considered both the Vietoris topology τV and the Fell topology τF.

Vietoris Fell

+ Familiar for topologists + F

d F

is Polish and com- pact + It is “the same” topology we put on K 0 1 d + Generates a standard Borel space

• Not metrizable if the ambient

space is not compact Coarser than Vietoris topology

Manlio Valenti On the descriptive complexity of Salem sets 15/20

Topology on F(Rd)

There is no “canonical” topology on the hyperspace F(Rd) of closed subsets of Rd. We considered both the Vietoris topology τV and the Fell topology τF.

Vietoris Fell

+ Familiar for topologists + F

d F

is Polish and com- pact + It is “the same” topology we put on K 0 1 d + Generates a standard Borel space

• Not metrizable if the ambient

space is not compact Coarser than Vietoris topology

Manlio Valenti On the descriptive complexity of Salem sets 15/20

Topology on F(Rd)

There is no “canonical” topology on the hyperspace F(Rd) of closed subsets of Rd. We considered both the Vietoris topology τV and the Fell topology τF.

Vietoris Fell

+ Familiar for topologists + F

d F

is Polish and com- pact + It is “the same” topology we put on K 0 1 d + Generates a standard Borel space

• Not metrizable if the ambient

space is not compact Coarser than Vietoris topology

Manlio Valenti On the descriptive complexity of Salem sets 15/20

Topology on F(Rd)

There is no “canonical” topology on the hyperspace F(Rd) of closed subsets of Rd. We considered both the Vietoris topology τV and the Fell topology τF.

Vietoris Fell

+ Familiar for topologists + F

d F

is Polish and com- pact + It is “the same” topology we put on K([0, 1]d) + Generates a standard Borel space

• Not metrizable if the ambient

space is not compact Coarser than Vietoris topology

Manlio Valenti On the descriptive complexity of Salem sets 15/20

Topology on F(Rd)

There is no “canonical” topology on the hyperspace F(Rd) of closed subsets of Rd. We considered both the Vietoris topology τV and the Fell topology τF.

Vietoris Fell

+ Familiar for topologists + F

d F

is Polish and com- pact + It is “the same” topology we put on K([0, 1]d) + Generates a standard Borel space

• Not metrizable if the ambient

space is not compact Coarser than Vietoris topology

Manlio Valenti On the descriptive complexity of Salem sets 15/20

Topology on F(Rd)

There is no “canonical” topology on the hyperspace F(Rd) of closed subsets of Rd. We considered both the Vietoris topology τV and the Fell topology τF.

Vietoris Fell

+ Familiar for topologists + F

d F

is Polish and com- pact + It is “the same” topology we put on K([0, 1]d) + Generates a standard Borel space

• Not metrizable if the ambient

space is not compact Coarser than Vietoris topology

Manlio Valenti On the descriptive complexity of Salem sets 15/20

Topology on F(Rd)

There is no “canonical” topology on the hyperspace F(Rd) of closed subsets of Rd. We considered both the Vietoris topology τV and the Fell topology τF.

Vietoris Fell

+ Familiar for topologists + (F(Rd), τF) is Polish and com- pact + It is “the same” topology we put on K([0, 1]d) + Generates a standard Borel space

• Not metrizable if the ambient

space is not compact Coarser than Vietoris topology

Manlio Valenti On the descriptive complexity of Salem sets 15/20

Topology on F(Rd)

There is no “canonical” topology on the hyperspace F(Rd) of closed subsets of Rd. We considered both the Vietoris topology τV and the Fell topology τF.

Vietoris Fell

+ Familiar for topologists + (F(Rd), τF) is Polish and com- pact + It is “the same” topology we put on K([0, 1]d) + Generates a standard Borel space

• Not metrizable if the ambient

space is not compact Coarser than Vietoris topology

Manlio Valenti On the descriptive complexity of Salem sets 15/20

Topology on F(Rd)

There is no “canonical” topology on the hyperspace F(Rd) of closed subsets of Rd. We considered both the Vietoris topology τV and the Fell topology τF.

Vietoris Fell

+ Familiar for topologists + (F(Rd), τF) is Polish and com- pact + It is “the same” topology we put on K([0, 1]d) + Generates a standard Borel space

• Not metrizable if the ambient

space is not compact ± Coarser than Vietoris topology

Manlio Valenti On the descriptive complexity of Salem sets 15/20

Stability of the Fourier dimension

Fourier dimension is (in general) not stable under countable unions and not inner-regular for compacts (Ekström et al [3]).

F n

An

n F An F A F K

K A and K is compact There is G

n Kn with F G

1 and

F Kn

0.

Theorem (Marcone, V.)

For every pointclass and every non-empty A

d , F A F K

K A is bounded and K

d

Manlio Valenti On the descriptive complexity of Salem sets 16/20

Stability of the Fourier dimension

Fourier dimension is (in general) not stable under countable unions and not inner-regular for compacts (Ekström et al [3]). dimF (∪

n

An ) ̸= sup

n dimF(An) F A F K

K A and K is compact There is G

n Kn with F G

1 and

F Kn

0.

Theorem (Marcone, V.)

For every pointclass and every non-empty A

d , F A F K

K A is bounded and K

d

Manlio Valenti On the descriptive complexity of Salem sets 16/20

Stability of the Fourier dimension

Fourier dimension is (in general) not stable under countable unions and not inner-regular for compacts (Ekström et al [3]). dimF (∪

n

An ) ̸= sup

n dimF(An)

dimF(A) ̸= sup{dimF(K) : K ⊂ A and K is compact} There is G

n Kn with F G

1 and

F Kn

0.

Theorem (Marcone, V.)

For every pointclass and every non-empty A

d , F A F K

K A is bounded and K

d

Manlio Valenti On the descriptive complexity of Salem sets 16/20

Stability of the Fourier dimension

Fourier dimension is (in general) not stable under countable unions and not inner-regular for compacts (Ekström et al [3]). dimF (∪

n

An ) ̸= sup

n dimF(An)

dimF(A) ̸= sup{dimF(K) : K ⊂ A and K is compact} There is G = ∪

n Kn with dimF(G) = 1 and dimF(Kn) = 0.

Theorem (Marcone, V.)

For every pointclass and every non-empty A

d , F A F K

K A is bounded and K

d

Manlio Valenti On the descriptive complexity of Salem sets 16/20

Stability of the Fourier dimension

Fourier dimension is (in general) not stable under countable unions and not inner-regular for compacts (Ekström et al [3]). dimF (∪

n

An ) ̸= sup

n dimF(An)

dimF(A) ̸= sup{dimF(K) : K ⊂ A and K is compact} There is G = ∪

n Kn with dimF(G) = 1 and dimF(Kn) = 0.

Theorem (Marcone, V.)

For every pointclass Γ and every non-empty A ∈ Γ(Rd), dimF(A) = sup{dimF(K) : K ⊊ A is bounded and K ∈ Γ(Rd)}.

Manlio Valenti On the descriptive complexity of Salem sets 16/20

Salem sets in Rd

Theorem (Marcone, V.)

Fix d ≥ 1. For every p < d the sets {A ∈ F(Rd) : dimH(A) > p}, {A ∈ F(Rd) : dimF(A) > p} are Σ0

2-complete. Moreover, for every q > 0 the sets

{A ∈ F(Rd) : dimH(A) ≥ q}, {A ∈ F(Rd) : dimF(A) ≥ q}, {A ∈ F(Rd) : A ∈ S (Rd)} are Π0

3-complete.

Manlio Valenti On the descriptive complexity of Salem sets 17/20

Efgectivizations

All the results obtained are actually efgective:

Theorem (Marcone, V.)

Let X be 0 1 d or

d for some d

1. A p F X 0 d A p is

2-complete

A p F X 0 d A p is

3-complete

A p F X 0 d

F A

p is

2-complete

A p F X 0 d

F A

p is

3-complete

A F X A X is

3-complete

Manlio Valenti On the descriptive complexity of Salem sets 18/20

Efgectivizations

All the results obtained are actually efgective:

Theorem (Marcone, V.)

Let X be [0, 1]d or Rd for some d ≥ 1.

• {(A, p) ∈ F(X) × [0, d) : dimH(A) > p} is Σ0

2-complete

• {(A, p) ∈ F(X) × (0, d] : dimH(A) ≥ p} is Π0

3-complete

• {(A, p) ∈ F(X) × [0, d) : dimF(A) > p} is Σ0

2-complete

• {(A, p) ∈ F(X) × (0, d] : dimF(A) ≥ p} is Π0

3-complete

• {A ∈ F(X) : A ∈ S (X)} is Π0

3-complete

Manlio Valenti On the descriptive complexity of Salem sets 18/20

Efgective measurability

f is called Σ0

k-measurable ifg preimages f−1(U) of open sets are

Σ0

k (relatively to dom(f)).

f is called efgectively

k-measurable if the preimage can be

uniformly computed from a name of U. Our results imply that the maps and

F are efgectively 3-measurable.

Manlio Valenti On the descriptive complexity of Salem sets 19/20

Efgective measurability

f is called Σ0

k-measurable ifg preimages f−1(U) of open sets are

Σ0

k (relatively to dom(f)).

f is called efgectively Σ0

k-measurable if the preimage can be

uniformly computed from a name of U. Our results imply that the maps and

F are efgectively 3-measurable.

Manlio Valenti On the descriptive complexity of Salem sets 19/20

Efgective measurability

f is called Σ0

k-measurable ifg preimages f−1(U) of open sets are

Σ0

k (relatively to dom(f)).

f is called efgectively Σ0

k-measurable if the preimage can be

uniformly computed from a name of U. Our results imply that the maps dimH and dimF are efgectively Σ0

3-measurable.

Manlio Valenti On the descriptive complexity of Salem sets 19/20

Weihrauch degree of dimF

Theorem (Brattka [1])

Let lim be the problem of fjnding the limit in the Baire space. f is efgectively Σ0

k+1-measurable ⇐

⇒ f ≤W lim[k]

Theorem (Marcone, V.)

F W lim 2

Manlio Valenti On the descriptive complexity of Salem sets 20/20

Weihrauch degree of dimF

Theorem (Brattka [1])

Let lim be the problem of fjnding the limit in the Baire space. f is efgectively Σ0

k+1-measurable ⇐

⇒ f ≤W lim[k]

Theorem (Marcone, V.)

dimF ≡W lim[2]

Manlio Valenti On the descriptive complexity of Salem sets 20/20

References

[1] Vasco Brattka, Efgective Borel measurability and reducibility of functions, Mathematical Logic Quarterly 51 (2005), no. 1, 19–44. [2] Vasco Brattka, Guido Gherardi, and Arno Pauly, Weihrauch Complexity in Computable Analysis, To appear, Available at https://arxiv.org/pdf/1707.03202v1. [3] Fredrik Ekström, Tomas Persson, and Jörg Schmeling, On the Fourier dimension and a modifjcation, Journal of Fractal Geometry 2 (2015), no. 3, 309–337. [4] Robert Fraser and Kyle Hambrook, Explicit Salem sets in Rn, Sep. 2019, Available at https://arxiv.org/abs/1909.04581. [5] Pertti Mattila, Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifjability, Cambridge University Press, 1995.

Manlio Valenti On the descriptive complexity of Salem sets 20/20