<latexit sha1_base64="GSrDr5l/UrxXNOIubAkLXq5ZcLs=">ACFHicbVC7SgNBFJ31bXxFLW0Gg2C17IqgZdDGUsGokIRwd3KjF2dnl5k7gixp/QS/wlYrO7G1t/Bf3KwpfJ3qcM59niTX5DiK3oOJyanpmdm5+drC4tLySn17cxl3ipsqUxn9iIBh5oMtphY40VuEdJE43lyfTjyz2/QOsrMKd/m2E3h0tCAFHAp9eqykwJfuUGRgtGUhTeg0TA1vSHfD4mHvXojCqMK8i+Jx6Qhxju1T86/Uz5tByjNDjXjqOcuwVYJqVxWOt4hzmoa7jEdkNpOi6RfXJUG5B5zJHK0kLSsRv3cUkDp3myZlZX3b28k/ue1PQ/2uwWZ3DMaNVrEpLFa5JSlMiKUfbLIDKPLUZKRCiwoyUJSpWiLzOrlXnEv7/S852wjgK45PdRvNgnMyc2BCbYlvEYk80xZE4Fi2hxJ14EI/iKbgPnoOX4PWrdCIY96yLHwjePgGrlJ9Y</latexit> <latexit sha1_base64="GSrDr5l/UrxXNOIubAkLXq5ZcLs=">ACFHicbVC7SgNBFJ31bXxFLW0Gg2C17IqgZdDGUsGokIRwd3KjF2dnl5k7gixp/QS/wlYrO7G1t/Bf3KwpfJ3qcM59niTX5DiK3oOJyanpmdm5+drC4tLySn17cxl3ipsqUxn9iIBh5oMtphY40VuEdJE43lyfTjyz2/QOsrMKd/m2E3h0tCAFHAp9eqykwJfuUGRgtGUhTeg0TA1vSHfD4mHvXojCqMK8i+Jx6Qhxju1T86/Uz5tByjNDjXjqOcuwVYJqVxWOt4hzmoa7jEdkNpOi6RfXJUG5B5zJHK0kLSsRv3cUkDp3myZlZX3b28k/ue1PQ/2uwWZ3DMaNVrEpLFa5JSlMiKUfbLIDKPLUZKRCiwoyUJSpWiLzOrlXnEv7/S852wjgK45PdRvNgnMyc2BCbYlvEYk80xZE4Fi2hxJ14EI/iKbgPnoOX4PWrdCIY96yLHwjePgGrlJ9Y</latexit> <latexit sha1_base64="GSrDr5l/UrxXNOIubAkLXq5ZcLs=">ACFHicbVC7SgNBFJ31bXxFLW0Gg2C17IqgZdDGUsGokIRwd3KjF2dnl5k7gixp/QS/wlYrO7G1t/Bf3KwpfJ3qcM59niTX5DiK3oOJyanpmdm5+drC4tLySn17cxl3ipsqUxn9iIBh5oMtphY40VuEdJE43lyfTjyz2/QOsrMKd/m2E3h0tCAFHAp9eqykwJfuUGRgtGUhTeg0TA1vSHfD4mHvXojCqMK8i+Jx6Qhxju1T86/Uz5tByjNDjXjqOcuwVYJqVxWOt4hzmoa7jEdkNpOi6RfXJUG5B5zJHK0kLSsRv3cUkDp3myZlZX3b28k/ue1PQ/2uwWZ3DMaNVrEpLFa5JSlMiKUfbLIDKPLUZKRCiwoyUJSpWiLzOrlXnEv7/S852wjgK45PdRvNgnMyc2BCbYlvEYk80xZE4Fi2hxJ14EI/iKbgPnoOX4PWrdCIY96yLHwjePgGrlJ9Y</latexit> <latexit sha1_base64="GSrDr5l/UrxXNOIubAkLXq5ZcLs=">ACFHicbVC7SgNBFJ31bXxFLW0Gg2C17IqgZdDGUsGokIRwd3KjF2dnl5k7gixp/QS/wlYrO7G1t/Bf3KwpfJ3qcM59niTX5DiK3oOJyanpmdm5+drC4tLySn17cxl3ipsqUxn9iIBh5oMtphY40VuEdJE43lyfTjyz2/QOsrMKd/m2E3h0tCAFHAp9eqykwJfuUGRgtGUhTeg0TA1vSHfD4mHvXojCqMK8i+Jx6Qhxju1T86/Uz5tByjNDjXjqOcuwVYJqVxWOt4hzmoa7jEdkNpOi6RfXJUG5B5zJHK0kLSsRv3cUkDp3myZlZX3b28k/ue1PQ/2uwWZ3DMaNVrEpLFa5JSlMiKUfbLIDKPLUZKRCiwoyUJSpWiLzOrlXnEv7/S852wjgK45PdRvNgnMyc2BCbYlvEYk80xZE4Fi2hxJ14EI/iKbgPnoOX4PWrdCIY96yLHwjePgGrlJ9Y</latexit> Department of Mathematics, Computer Science, Physics University of Udine On the descriptive complexity of Salem sets Manlio Valenti manlio . valenti @ uniud . it Joint work with Alberto Marcone CCA Sep, 11, 2020
Question During the IMS Graduate Summer School in Logic in 2018, Slaman asked: “What is the descriptive complexity of the family Manlio Valenti On the descriptive complexity of Salem sets 1/20 of closed Salem sets in [ 0 , 1 ] ?”
Hausdorfg dimension d On the descriptive complexity of Salem sets Manlio Valenti In other words, it coincides with the capacitary dimension. cr s B x r 0 r x Standard notion in geometric measure theory. 0 c A s A can be written as d Using Frostman’s lemma, the Hausdorfg dimension of A 2/20
Hausdorfg dimension Standard notion in geometric measure theory. can be written as In other words, it coincides with the capacitary dimension. Manlio Valenti On the descriptive complexity of Salem sets 2/20 Using Frostman’s lemma, the Hausdorfg dimension of A ∈ B ( R d ) dim H ( A ) = sup { s : ( ∃ µ ∈ P ( A )) ( ∃ c > 0 ) ( ∀ x ∈ R d ) ( ∀ r > 0 ) ( µ ( B ( x , r )) ≤ cr s ) }
x d d is defjned as F A 0 d On the descriptive complexity of Salem sets Manlio Valenti s 2 c x x d x 0 c A Fourier dimension s Let The Fourier dimension of A x i e defjned as d : Fourier transform of be a fjnite Borel measure. 3/20
d is defjned as F A Fourier dimension c On the descriptive complexity of Salem sets Manlio Valenti s 2 c x x d x 0 0 d A s The Fourier dimension of A 3/20 Let µ be a fjnite Borel measure. µ : R d → C defjned as Fourier transform of µ : � ∫ e − i ξ · x d µ ( x ) µ ( ξ ) := �
Manlio Valenti Fourier dimension On the descriptive complexity of Salem sets 3/20 Let µ be a fjnite Borel measure. µ : R d → C defjned as Fourier transform of µ : � ∫ e − i ξ · x d µ ( x ) � µ ( ξ ) := The Fourier dimension of A ⊂ R d is defjned as dim F ( A ) := sup { s ∈ [ 0 , d ] : ( ∃ µ ∈ P ( A )) ( ∃ c > 0 ) ( ∀ x ∈ R d ) µ ( x ) | ≤ c | x | − s / 2 ) } ( | �
F A is called Salem set . Salem sets Proposition (Folklore?) This shows that the Fourier dimension can be used to obtain lower bounds for the Hausdorfg dimension. A set A s.t. A Manlio Valenti On the descriptive complexity of Salem sets 4/20 For every A ∈ B ( R d ) we have dim F ( A ) ≤ dim H ( A )
Salem sets Proposition (Folklore?) This shows that the Fourier dimension can be used to obtain lower bounds for the Hausdorfg dimension. Manlio Valenti On the descriptive complexity of Salem sets 4/20 For every A ∈ B ( R d ) we have dim F ( A ) ≤ dim H ( A ) A set A s.t. dim H ( A ) = dim F ( A ) is called Salem set .
Salem sets For every On the descriptive complexity of Salem sets Manlio Valenti 2 2 -well approximable x is 0 1 x 0 Theorem (Jarník 1928, Besicovitch 1934, Kaufmann 1981) Examples: Classic example: Jarník’s fractal. Deterministic (non-trivial) Salem sets are rare. 0 F and 3 2 Cantor middle-third set is not Salem: it has , 0 1 are Salem subsets of 0 1 . 5/20
Salem sets 0 On the descriptive complexity of Salem sets Manlio Valenti 2 2 -well approximable x is 0 1 x For every Theorem (Jarník 1928, Besicovitch 1934, Kaufmann 1981) Classic example: Jarník’s fractal. Deterministic (non-trivial) Salem sets are rare. 0 F and 3 2 Cantor middle-third set is not Salem: it has 5/20 Examples: ∅ , [ 0 , 1 ] are Salem subsets of [ 0 , 1 ] .
Salem sets 0 On the descriptive complexity of Salem sets Manlio Valenti 2 2 -well approximable x is 0 1 x For every Theorem (Jarník 1928, Besicovitch 1934, Kaufmann 1981) Classic example: Jarník’s fractal. Deterministic (non-trivial) Salem sets are rare. and Cantor middle-third set is not Salem: it has 5/20 Examples: ∅ , [ 0 , 1 ] are Salem subsets of [ 0 , 1 ] . dim H = log( 2 ) dim F = 0 log( 3 )
Salem sets 0 On the descriptive complexity of Salem sets Manlio Valenti 2 2 -well approximable x is 0 1 x For every Theorem (Jarník 1928, Besicovitch 1934, Kaufmann 1981) Classic example: Jarník’s fractal. Deterministic (non-trivial) Salem sets are rare. and Cantor middle-third set is not Salem: it has 5/20 Examples: ∅ , [ 0 , 1 ] are Salem subsets of [ 0 , 1 ] . dim H = log( 2 ) dim F = 0 log( 3 )
Salem sets Deterministic (non-trivial) Salem sets are rare. On the descriptive complexity of Salem sets Manlio Valenti 2 Theorem (Jarník 1928, Besicovitch 1934, Kaufmann 1981) Classic example: Jarník’s fractal. Cantor middle-third set is not Salem: it has and 5/20 Examples: ∅ , [ 0 , 1 ] are Salem subsets of [ 0 , 1 ] . dim H = log( 2 ) dim F = 0 log( 3 ) For every α ≥ 0 dim( { x ∈ [ 0 , 1 ] : x is α -well approximable } ) = 2 + α
W B ) if there is a W B . Wadge reducibility B is called On the descriptive complexity of Salem sets Manlio Valenti Y -hard and B -complete if it is B is called , A 2 -hard if, for every A B We will locate the family of closed Salem sets in the Borel hierarchy f x A x Y s.t. X continuous map f Y ( A X is Wadge reducible to B A Let X and Y be Polish spaces. 6/20
W B . Wadge reducibility , A On the descriptive complexity of Salem sets Manlio Valenti Y -hard and B -complete if it is B is called 2 We will locate the family of closed Salem sets in the Borel hierarchy -hard if, for every A B is called Let X and Y be Polish spaces. 6/20 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
Wadge reducibility We will locate the family of closed Salem sets in the Borel hierarchy Let X and Y be Polish spaces. Manlio Valenti On the descriptive complexity of Salem sets 6/20 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 ( 2 N ) , A ≤ W B . B is called Γ -complete if it is Γ -hard and B ∈ Γ ( Y )
F A F A 0 1 p is 0 2 ; A p K 0 1 7/20 K 0 1 p is 0 3 . The proof relies on the compactness of the ambient space 0 1 . Manlio Valenti On the descriptive complexity of Salem sets 0 1 A p 2 ; 3 ; K 0 1 0 1 A p is 0 A p Lemma (Marcone, Reimann, Slaman, V.) K 0 1 0 1 A p is 0 A p Salem sets in [ 0 , 1 ] Work in K ([ 0 , 1 ]) : the hyperspace of compact subsets of [ 0 , 1 ] .
Lemma (Marcone, Reimann, Slaman, V.) 2 ; 3 ; 2 ; 3 . 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 ] . • { ( A , p ) ∈ K ([ 0 , 1 ]) × [ 0 , 1 ] : dim H ( A ) > p } is Σ 0 • { ( A , p ) ∈ K ([ 0 , 1 ]) × [ 0 , 1 ] : dim H ( A ) ≥ p } is Π 0 • { ( A , p ) ∈ K ([ 0 , 1 ]) × [ 0 , 1 ] : dim F ( A ) > p } is Σ 0 • { ( A , p ) ∈ K ([ 0 , 1 ]) × [ 0 , 1 ] : dim F ( A ) ≥ p } is Π 0 The proof relies on the compactness of the ambient space [ 0 , 1 ] .
p Lemma (Marcone, Reimann, Slaman, V.) 0 2 -complete. Manlio Valenti On the descriptive complexity of Salem sets 8/20 Salem sets in [ 0 , 1 ] For every p ∈ [ 0 , 1 ] there is a continuous (in fact computable) map f p : 2 N → S ([ 0 , 1 ]) s.t. { if x ∈ Q 2 dim( f p ( x )) = if x / ∈ Q 2 where Q 2 = { x ∈ 2 N : ( ∀ ∞ n )( x ( n ) = 0 ) } is Σ 0
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