On continuous functions definable in expansions of the ordered real - - PowerPoint PPT Presentation

On continuous functions definable in expansions of the

• rdered real additive group

Philipp Hieronymi

University of Illinois at Urbana-Champaign

Bedlewo July 2017

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 1 / 14

Throughout this talk, we will consider expansions R of (R, <, +). Motivating question. Can we classify such expansions according to the geometric/topological complexity of its definable sets (definable functions)?

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 2 / 14

Throughout this talk, we will consider expansions R of (R, <, +). Motivating question. Can we classify such expansions according to the geometric/topological complexity of its definable sets (definable functions)? A disclaimer. Two perspectives on first-order expansions of (R, <, +):

1 as a concrete collection of (definable) subsets of Rn, 2 in terms of its theory, up to bi-interpretability.

For this talk, we take perspective 1.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 2 / 14

Throughout this talk, we will consider expansions R of (R, <, +). Motivating question. Can we classify such expansions according to the geometric/topological complexity of its definable sets (definable functions)? A disclaimer. Two perspectives on first-order expansions of (R, <, +):

1 as a concrete collection of (definable) subsets of Rn, 2 in terms of its theory, up to bi-interpretability.

For this talk, we take perspective 1.

• References. Unless otherwise stated (or said) the results are from the

following three papers: H.-Walsberg, ‘Interpreting the monadic second order theory of one successor in expansions of the real line’, Israel J. to appear, Fornasiero-H.-Walsberg, ‘How to avoid a compact set’, Preprint H.-Walsberg, ‘On continuous functions definable in expansions of the

• rdered real additive group’, Preprint soon

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 2 / 14

An ω-orderable set is a definable set that admits a definable ordering with order type ω. We say that such a set is dense if it is dense in some

• pen subinterval of R.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 3 / 14

An ω-orderable set is a definable set that admits a definable ordering with order type ω. We say that such a set is dense if it is dense in some

• pen subinterval of R.

Trivial Trichotomy. An expansion R of (R, <, +) satisfies exactly one of the following three conditions: (A) R does not define a dense ω-orderable set. (B) R defines a dense ω-orderable, but avoids a compact set. (C) R defines every compact set.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 3 / 14

An ω-orderable set is a definable set that admits a definable ordering with order type ω. We say that such a set is dense if it is dense in some

• pen subinterval of R.

Trivial Trichotomy. An expansion R of (R, <, +) satisfies exactly one of the following three conditions: (A) R does not define a dense ω-orderable set. (B) R defines a dense ω-orderable, but avoids a compact set. (C) R defines every compact set. Observation: Type C ⇒ No model-theoretic tameness.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 3 / 14

An ω-orderable set is a definable set that admits a definable ordering with order type ω. We say that such a set is dense if it is dense in some

• pen subinterval of R.

Trivial Trichotomy. An expansion R of (R, <, +) satisfies exactly one of the following three conditions: (A) R does not define a dense ω-orderable set. (B) R defines a dense ω-orderable, but avoids a compact set. (C) R defines every compact set. DST Observation: Type C ⇒ No model-theoretic tameness.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 3 / 14

An ω-orderable set is a definable set that admits a definable ordering with order type ω. We say that such a set is dense if it is dense in some

• pen subinterval of R.

Trivial Trichotomy. An expansion R of (R, <, +) satisfies exactly one of the following three conditions: (A) R does not define a dense ω-orderable set. (B) R defines a dense ω-orderable, but avoids a compact set. (C) R defines every compact set. DST Observation: Type C ⇒ No model-theoretic tameness. Observation: O-minimality ⇒ Type A.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 3 / 14

An ω-orderable set is a definable set that admits a definable ordering with order type ω. We say that such a set is dense if it is dense in some

• pen subinterval of R.

Trivial Trichotomy. An expansion R of (R, <, +) satisfies exactly one of the following three conditions: (A) R does not define a dense ω-orderable set. (B) R defines a dense ω-orderable, but avoids a compact set. (C) R defines every compact set. DST Observation: Type C ⇒ No model-theoretic tameness. Observation: O-minimality ⇒ Type A. Observation: NTP2 ⇒ Type A.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 3 / 14

An ω-orderable set is a definable set that admits a definable ordering with order type ω. We say that such a set is dense if it is dense in some

• pen subinterval of R.

Trivial Trichotomy. An expansion R of (R, <, +) satisfies exactly one of the following three conditions: (A) R does not define a dense ω-orderable set. Tame geometry (B) R defines a dense ω-orderable, but avoids a compact set. (C) R defines every compact set. DST Observation: Type C ⇒ No model-theoretic tameness. Observation: O-minimality ⇒ Type A. Observation: NTP2 ⇒ Type A.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 3 / 14

An ω-orderable set is a definable set that admits a definable ordering with order type ω. We say that such a set is dense if it is dense in some

• pen subinterval of R.

Trivial Trichotomy. An expansion R of (R, <, +) satisfies exactly one of the following three conditions: (A) R does not define a dense ω-orderable set. Tame geometry (B) R defines a dense ω-orderable, but avoids a compact set. (C) R defines every compact set. DST Observation: Type C ⇒ No model-theoretic tameness. Observation: O-minimality ⇒ Type A. Observation: NTP2 ⇒ Type A. Observation: Type B interprets (N, P(N), +1, ∈), can be decidable.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 3 / 14

An ω-orderable set is a definable set that admits a definable ordering with order type ω. We say that such a set is dense if it is dense in some

• pen subinterval of R.

Trivial Trichotomy. An expansion R of (R, <, +) satisfies exactly one of the following three conditions: (A) R does not define a dense ω-orderable set. Tame geometry (B) R defines a dense ω-orderable, but avoids a compact set. Automata (C) R defines every compact set. DST Observation: Type C ⇒ No model-theoretic tameness. Observation: O-minimality ⇒ Type A. Observation: NTP2 ⇒ Type A. Observation: Type B interprets (N, P(N), +1, ∈), can be decidable.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 3 / 14

An ω-orderable set is a definable set that admits a definable ordering with order type ω. We say that such a set is dense if it is dense in some

• pen subinterval of R.

Trivial Trichotomy. An expansion R of (R, <, +) satisfies exactly one of the following three conditions: (A) R does not define a dense ω-orderable set. Tame geometry (B) R defines a dense ω-orderable, but avoids a compact set. Automata (C) R defines every compact set. DST Observation: Type C ⇒ No model-theoretic tameness. Observation: O-minimality ⇒ Type A. Observation: NTP2 ⇒ Type A. Observation: Type B interprets (N, P(N), +1, ∈), can be decidable. Observation: Type A can interpret (R, <, +, ·, Z).

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 3 / 14

An ω-orderable set is a definable set that admits a definable ordering with order type ω. We say that such a set is dense if it is dense in some

• pen subinterval of R.

Trivial Trichotomy. An expansion R of (R, <, +) satisfies exactly one of the following three conditions: (A) R does not define a dense ω-orderable set. Tame geometry (B) R defines a dense ω-orderable, but avoids a compact set. Automata (C) R defines every compact set. DST Observation: Type C ⇒ No model-theoretic tameness. Observation: O-minimality ⇒ Type A. Observation: NTP2 ⇒ Type A. Observation: Type B interprets (N, P(N), +1, ∈), can be decidable. Observation: Type A can interpret (R, <, +, ·, Z). ‘What about decidability of the theory? Just as biological taxonomy does not tell us whether a species is tasty, the classificaton here does not deal with decidability.’ - Saharon Shelah

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 3 / 14

• Definition. An infinite definable subset of Rn is ω-orderable if it admits a

definable ordering with order type ω. We say that such a set is dense if it is dense in some open subinterval of R.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 4 / 14

• Definition. An infinite definable subset of Rn is ω-orderable if it admits a

definable ordering with order type ω. We say that such a set is dense if it is dense in some open subinterval of R. Observations. If D is ω-orderable, so is Dk.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 4 / 14

• Definition. An infinite definable subset of Rn is ω-orderable if it admits a

definable ordering with order type ω. We say that such a set is dense if it is dense in some open subinterval of R. Observations. If D is ω-orderable, so is Dk. If D is ω-orderable and f : D → R is definable, then f (D) is ω-orderable (if f (D) is infinite).

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 4 / 14

• Definition. An infinite definable subset of Rn is ω-orderable if it admits a

definable ordering with order type ω. We say that such a set is dense if it is dense in some open subinterval of R. Observations. If D is ω-orderable, so is Dk. If D is ω-orderable and f : D → R is definable, then f (D) is ω-orderable (if f (D) is infinite). If D ⊆ R is infinite, closed and discrete, then D is ω-orderable.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 4 / 14

• Definition. An infinite definable subset of Rn is ω-orderable if it admits a

definable ordering with order type ω. We say that such a set is dense if it is dense in some open subinterval of R. Observations. If D is ω-orderable, so is Dk. If D is ω-orderable and f : D → R is definable, then f (D) is ω-orderable (if f (D) is infinite). If D ⊆ R is infinite, closed and discrete, then D is ω-orderable. If D ⊆ R is closed and discrete set and f : Dn → R is such that f (Dn) is somewhere dense, then f (Dn) is a dense ω-orderable set.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 4 / 14

• Definition. An infinite definable subset of Rn is ω-orderable if it admits a

definable ordering with order type ω. We say that such a set is dense if it is dense in some open subinterval of R. Observations. If D is ω-orderable, so is Dk. If D is ω-orderable and f : D → R is definable, then f (D) is ω-orderable (if f (D) is infinite). If D ⊆ R is infinite, closed and discrete, then D is ω-orderable. If D ⊆ R is closed and discrete set and f : Dn → R is such that f (Dn) is somewhere dense, then f (Dn) is a dense ω-orderable set.

• Theorem. Let R be an expansion of the real field. If R defines a dense

ω-orderable set, then R defines Z

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 4 / 14

• Definition. An infinite definable subset of Rn is ω-orderable if it admits a

definable ordering with order type ω. We say that such a set is dense if it is dense in some open subinterval of R. Observations. If D is ω-orderable, so is Dk. If D is ω-orderable and f : D → R is definable, then f (D) is ω-orderable (if f (D) is infinite). If D ⊆ R is infinite, closed and discrete, then D is ω-orderable. If D ⊆ R is closed and discrete set and f : Dn → R is such that f (Dn) is somewhere dense, then f (Dn) is a dense ω-orderable set.

• Theorem. Let R be an expansion of the real field. If R defines a dense

ω-orderable set, then R defines Z and hence R defines every closed set.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 4 / 14

• Definition. An infinite definable subset of Rn is ω-orderable if it admits a

definable ordering with order type ω. We say that such a set is dense if it is dense in some open subinterval of R. Observations. If D is ω-orderable, so is Dk. If D is ω-orderable and f : D → R is definable, then f (D) is ω-orderable (if f (D) is infinite). If D ⊆ R is infinite, closed and discrete, then D is ω-orderable. If D ⊆ R is closed and discrete set and f : Dn → R is such that f (Dn) is somewhere dense, then f (Dn) is a dense ω-orderable set.

• Theorem. Let R be an expansion of the real field. If R defines a dense

ω-orderable set, then R defines Z and hence R defines every closed set.

• Cor. If R expands the real field, then R can not be type B.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 4 / 14

For r ∈ N≥2, consider a ternary predicate Vr(x, u, k) that holds if and only if u is an integer power of r, k ∈ {0, . . . , r − 1}, and the digit of some base r representation of x in the position corresponding to u is k. Set Tr := (R, <, +, Vr).

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 5 / 14

For r ∈ N≥2, consider a ternary predicate Vr(x, u, k) that holds if and only if u is an integer power of r, k ∈ {0, . . . , r − 1}, and the digit of some base r representation of x in the position corresponding to u is k. Set Tr := (R, <, +, Vr). Observation: Tr is bi-interpretable with (N, P(N), +1, ∈).

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 5 / 14

For r ∈ N≥2, consider a ternary predicate Vr(x, u, k) that holds if and only if u is an integer power of r, k ∈ {0, . . . , r − 1}, and the digit of some base r representation of x in the position corresponding to u is k. Set Tr := (R, <, +, Vr). Observation: Tr is bi-interpretable with (N, P(N), +1, ∈). Let Dr be the set of numbers in [0, 1) admitting a finite base r expansion.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 5 / 14

For r ∈ N≥2, consider a ternary predicate Vr(x, u, k) that holds if and only if u is an integer power of r, k ∈ {0, . . . , r − 1}, and the digit of some base r representation of x in the position corresponding to u is k. Set Tr := (R, <, +, Vr). Observation: Tr is bi-interpretable with (N, P(N), +1, ∈). Let Dr be the set of numbers in [0, 1) admitting a finite base r expansion. Define τr : Dr → r−N>0 so that τr(d) is the least u ∈ r−N>0 appearing with nonzero coefficient in the finite base r expansion of d.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 5 / 14

For r ∈ N≥2, consider a ternary predicate Vr(x, u, k) that holds if and only if u is an integer power of r, k ∈ {0, . . . , r − 1}, and the digit of some base r representation of x in the position corresponding to u is k. Set Tr := (R, <, +, Vr). Observation: Tr is bi-interpretable with (N, P(N), +1, ∈). Let Dr be the set of numbers in [0, 1) admitting a finite base r expansion. Define τr : Dr → r−N>0 so that τr(d) is the least u ∈ r−N>0 appearing with nonzero coefficient in the finite base r expansion of d. For d, e ∈ Dr, let d ≺r e ⇐ ⇒ τr(d) > τr(e) or (τr(d) = τr(e) and d < e) .

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 5 / 14

For r ∈ N≥2, consider a ternary predicate Vr(x, u, k) that holds if and only if u is an integer power of r, k ∈ {0, . . . , r − 1}, and the digit of some base r representation of x in the position corresponding to u is k. Set Tr := (R, <, +, Vr). Observation: Tr is bi-interpretable with (N, P(N), +1, ∈). Let Dr be the set of numbers in [0, 1) admitting a finite base r expansion. Define τr : Dr → r−N>0 so that τr(d) is the least u ∈ r−N>0 appearing with nonzero coefficient in the finite base r expansion of d. For d, e ∈ Dr, let d ≺r e ⇐ ⇒ τr(d) > τr(e) or (τr(d) = τr(e) and d < e) . Observation: (Dr, ≺) is definable in Tr and has order type ω.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 5 / 14

Two more natural (non-obvious) examples of type B structures: (R, <, +, C), where C is the middle-thirds Cantor set C,

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 6 / 14

Two more natural (non-obvious) examples of type B structures: (R, <, +, C), where C is the middle-thirds Cantor set C, (R, <, +, Z, x → αx), where α is irrational and quadratic.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 6 / 14

Two more natural (non-obvious) examples of type B structures: (R, <, +, C), where C is the middle-thirds Cantor set C, (R, <, +, Z, x → αx), where α is irrational and quadratic. Both structures are bi-interpretable with (N, P(N), +1, ∈). Open question. Is this true for every type B structure?

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 6 / 14

Two more natural (non-obvious) examples of type B structures: (R, <, +, C), where C is the middle-thirds Cantor set C, (R, <, +, Z, x → αx), where α is irrational and quadratic. Both structures are bi-interpretable with (N, P(N), +1, ∈). Open question. Is this true for every type B structure? It is every hard to expand type B structures without becoming type C:

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 6 / 14

Two more natural (non-obvious) examples of type B structures: (R, <, +, C), where C is the middle-thirds Cantor set C, (R, <, +, Z, x → αx), where α is irrational and quadratic. Both structures are bi-interpretable with (N, P(N), +1, ∈). Open question. Is this true for every type B structure? It is every hard to expand type B structures without becoming type C: with α and C as above, (R, <, +, x → αx, C) is type C. with α irrational and non-quadratic, (R, <, +, Z, x → αx) is type C.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 6 / 14

Trivial Trichotomy. An expansion R of (R, <, +) satisfies exactly one of the following three conditions: (A) R does not define a dense ω-orderable set. (B) R defines a dense ω-orderable, but avoids a compact set. (C) R defines every compact set. Type C. Every continuous function f : [0, 1] → R is definable.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 7 / 14

Trivial Trichotomy. An expansion R of (R, <, +) satisfies exactly one of the following three conditions: (A) R does not define a dense ω-orderable set. (B) R defines a dense ω-orderable, but avoids a compact set. (C) R defines every compact set. Type C. Every continuous function f : [0, 1] → R is definable. Type A. Let f : [0, 1] → R be a definable continuous function on an open interval I. Then for every k ∈ N there is a definable open dense U ⊆ I on which f is C k.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 7 / 14

Trivial Trichotomy. An expansion R of (R, <, +) satisfies exactly one of the following three conditions: (A) R does not define a dense ω-orderable set. (B) R defines a dense ω-orderable, but avoids a compact set. (C) R defines every compact set. Type C. Every continuous function f : [0, 1] → R is definable. Type A. Let f : [0, 1] → R be a definable continuous function on an open interval I. Then for every k ∈ N there is a definable open dense U ⊆ I on which f is C k. Type B. Every definable C 2 function f : [0, 1] → R is affine.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 7 / 14

Trivial Trichotomy. An expansion R of (R, <, +) satisfies exactly one of the following three conditions: (A) R does not define a dense ω-orderable set. (B) R defines a dense ω-orderable, but avoids a compact set. (C) R defines every compact set. Type C. Every continuous function f : [0, 1] → R is definable. Type A. Let f : [0, 1] → R be a definable continuous function on an open interval I. Then for every k ∈ N there is a definable open dense U ⊆ I on which f is C k. Uses idea of Laskowski and Steinhorn (to use a Theorem of Boas-Widder.) Type B. Every definable C 2 function f : [0, 1] → R is affine.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 7 / 14

Trivial Trichotomy. An expansion R of (R, <, +) satisfies exactly one of the following three conditions: (A) R does not define a dense ω-orderable set. (B) R defines a dense ω-orderable, but avoids a compact set. (C) R defines every compact set. Type C. Every continuous function f : [0, 1] → R is definable. Type A. Let f : [0, 1] → R be a definable continuous function on an open interval I. Then for every k ∈ N there is a definable open dense U ⊆ I on which f is C k. Uses idea of Laskowski and Steinhorn (to use a Theorem of Boas-Widder.) Type B. Every definable C 2 function f : [0, 1] → R is affine. Use results of H. and Tychonievich.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 7 / 14

An application to generic functions

Let C ∞([0, 1]) be the space of smooth functions with the topology induced by the semi-norms f → maxt∈[0,1] |f (j)(t)| for j ∈ N.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 8 / 14

An application to generic functions

Let C ∞([0, 1]) be the space of smooth functions with the topology induced by the semi-norms f → maxt∈[0,1] |f (j)(t)| for j ∈ N. Grigoriev 05/LeGal 09. The set of all f ∈ C ∞([0, 1]) such that (R, <, +, f ) is o-minimal, is co-meager.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 8 / 14

An application to generic functions

Let C ∞([0, 1]) be the space of smooth functions with the topology induced by the semi-norms f → maxt∈[0,1] |f (j)(t)| for j ∈ N. Grigoriev 05/LeGal 09. The set of all f ∈ C ∞([0, 1]) such that (R, <, +, f ) is o-minimal, is co-meager. Let C k([0, 1]) be the space of all C k functions [0, 1] → R equipped with the topology induced by the semi-norms f → maxt∈[0,1] |f (j)(t)| for 0 ≤ j ≤ k.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 8 / 14

An application to generic functions

Let C ∞([0, 1]) be the space of smooth functions with the topology induced by the semi-norms f → maxt∈[0,1] |f (j)(t)| for j ∈ N. Grigoriev 05/LeGal 09. The set of all f ∈ C ∞([0, 1]) such that (R, <, +, f ) is o-minimal, is co-meager. Let C k([0, 1]) be the space of all C k functions [0, 1] → R equipped with the topology induced by the semi-norms f → maxt∈[0,1] |f (j)(t)| for 0 ≤ j ≤ k.

• Theorem. The set of all f ∈ C k([0, 1]) such that (R, <, +, f ) is type C, is

co-meager in C k([0, 1]).

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 8 / 14

An application to generic functions

Let C ∞([0, 1]) be the space of smooth functions with the topology induced by the semi-norms f → maxt∈[0,1] |f (j)(t)| for j ∈ N. Grigoriev 05/LeGal 09. The set of all f ∈ C ∞([0, 1]) such that (R, <, +, f ) is o-minimal, is co-meager. Let C k([0, 1]) be the space of all C k functions [0, 1] → R equipped with the topology induced by the semi-norms f → maxt∈[0,1] |f (j)(t)| for 0 ≤ j ≤ k.

• Theorem. The set of all f ∈ C k([0, 1]) such that (R, <, +, f ) is type C, is

co-meager in C k([0, 1]).

• Proof. It is well-known that the set of somewhere (k + 1)-differentiable

functions in C k([0, 1]) is meager. This rules out type A. It is left to show that the set of all f ∈ C k([0, 1]) such that (R, <, +, f ) is type B is

• meager. When k ≥ 2, these are just affine functions.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 8 / 14

• Definition. We say that R is of field-type if there is an interval I ⊆ R

together with definable functions ⊕, ⊗ : I 2 → I such that (I, <, ⊕, ⊗) is an ordered field isomorphic to (R, <, +, ·).

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 9 / 14

• Definition. We say that R is of field-type if there is an interval I ⊆ R

together with definable functions ⊕, ⊗ : I 2 → I such that (I, <, ⊕, ⊗) is an ordered field isomorphic to (R, <, +, ·). Marker-Pillay-Peterzil 92. Suppose R is o-minimal. If R defines a non-affine C 1 function [0, 1] → R, then R is of field-type.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 9 / 14

• Definition. We say that R is of field-type if there is an interval I ⊆ R

together with definable functions ⊕, ⊗ : I 2 → I such that (I, <, ⊕, ⊗) is an ordered field isomorphic to (R, <, +, ·). Marker-Pillay-Peterzil 92. Suppose R is o-minimal. If R defines a non-affine C 1 function [0, 1] → R, then R is of field-type.

• Theorem. Suppose R defines a non-affine C 1 function [0, 1] → R. Then
• ne of the following holds:

1 R is of field-type. 2 R interprets (R, <, +, ·, Z). Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 9 / 14

• Definition. We say that R is of field-type if there is an interval I ⊆ R

together with definable functions ⊕, ⊗ : I 2 → I such that (I, <, ⊕, ⊗) is an ordered field isomorphic to (R, <, +, ·). Marker-Pillay-Peterzil 92. Suppose R is o-minimal. If R defines a non-affine C 1 function [0, 1] → R, then R is of field-type.

• Theorem. Suppose R defines a non-affine C 1 function [0, 1] → R. Then
• ne of the following holds:

1 R is of field-type. In particular, when f is C 2. 2 R interprets (R, <, +, ·, Z). Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 9 / 14

• Definition. We say that R is of field-type if there is an interval I ⊆ R

together with definable functions ⊕, ⊗ : I 2 → I such that (I, <, ⊕, ⊗) is an ordered field isomorphic to (R, <, +, ·). Marker-Pillay-Peterzil 92. Suppose R is o-minimal. If R defines a non-affine C 1 function [0, 1] → R, then R is of field-type.

• Theorem. Suppose R defines a non-affine C 1 function [0, 1] → R. Then
• ne of the following holds:

1 R is of field-type. In particular, when f is C 2. 2 R interprets (R, <, +, ·, Z).

• Cor. Suppose that R is type A. Then one of the following holds:

1 R is of field-type. 2 For every continuous definable f : [0, 1] → R there is an open dense

U ⊆ [0, 1] such that f is affine on each connected component of U.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 9 / 14

• Definition. We say that R is of field-type if there is an interval I ⊆ R

together with definable functions ⊕, ⊗ : I 2 → I such that (I, <, ⊕, ⊗) is an ordered field isomorphic to (R, <, +, ·). Marker-Pillay-Peterzil 92. Suppose R is o-minimal. If R defines a non-affine C 1 function [0, 1] → R, then R is of field-type.

• Theorem. Suppose R defines a non-affine C 1 function [0, 1] → R. Then
• ne of the following holds:

1 R is of field-type. In particular, when f is C 2. 2 R interprets (R, <, +, ·, Z).

• Cor. Suppose that R is type A. Then one of the following holds:

1 R is of field-type. 2 For every continuous definable f : [0, 1] → R there is an open dense

U ⊆ [0, 1] such that f is affine on each connected component of U.

• Cor. Suppose that R is type B and does not interpret (R, <, +, ·, Z).

Then every definable C 1 function f : [0, 1] → R is affine.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 9 / 14

Step 1. Let a, b ⊆ R and let f : [a, b] → R be a definable non-affine C 1 function.

1 If f ′ is strictly increasing or strictly decreasing on some open

subinterval of I, then R is of field-type.

2 If f ′ is not strictly increasing or strictly decreasing on any open

subinterval of I, then R defines a nowhere dense F ⊆ R and ⊕, ⊗ : F 2 → F be definable functions such that (F, <, ⊕, ⊗) is an

• rdered field isomorphic to (R, <, +, ·).

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 10 / 14

Step 1. Let a, b ⊆ R and let f : [a, b] → R be a definable non-affine C 1 function.

1 If f ′ is strictly increasing or strictly decreasing on some open

subinterval of I, then R is of field-type.

2 If f ′ is not strictly increasing or strictly decreasing on any open

subinterval of I, then R defines a nowhere dense F ⊆ R and ⊕, ⊗ : F 2 → F be definable functions such that (F, <, ⊕, ⊗) is an

• rdered field isomorphic to (R, <, +, ·).

Step 2. Let F ⊆ R be nowhere dense and ⊕, ⊗ : F 2 → F be definable functions such that (F, <, ⊕, ⊗) is an ordered field isomorphic to (R, <, +, ·). Then there is a definable set Z ⊆ F such that (F, <, ⊕, ⊗, Z) is isomorphic to (R, <, +, ·, Z).

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 10 / 14

Let F ⊆ R be nowhere dense and ⊕, ⊗ : F 2 → F be definable functions such that (F, <, ⊕, ⊗) is an ordered field isomorphic to (R, <, +, ·). Then there is a definable set Z ⊆ F such that (F, <, ⊕, ⊗, Z) is isomorphic to (R, <, +, ·, Z). Proof:

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 11 / 14

Let F ⊆ R be nowhere dense and ⊕, ⊗ : F 2 → F be definable functions such that (F, <, ⊕, ⊗) is an ordered field isomorphic to (R, <, +, ·). Then there is a definable set Z ⊆ F such that (F, <, ⊕, ⊗, Z) is isomorphic to (R, <, +, ·, Z). Proof: Let ι : (F, <, ⊕, ⊗) → (R, <, +, ·) be an isomorphism. Enough to show that ι−1(Z) is definable, because then ι : (F, <, ⊕, ⊗, Z) → (R, <, +, ·, Z).

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 11 / 14

Let F ⊆ R be nowhere dense and ⊕, ⊗ : F 2 → F be definable functions such that (F, <, ⊕, ⊗) is an ordered field isomorphic to (R, <, +, ·). Then there is a definable set Z ⊆ F such that (F, <, ⊕, ⊗, Z) is isomorphic to (R, <, +, ·, Z). Proof: Let ι : (F, <, ⊕, ⊗) → (R, <, +, ·) be an isomorphism. Enough to show that ι−1(Z) is definable, because then ι : (F, <, ⊕, ⊗, Z) → (R, <, +, ·, Z). Let U be the complement of the closure of F. Let D ⊆ F be the definable set of endpoints of bounded connected components of U that lie in F.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 11 / 14

Let F ⊆ R be nowhere dense and ⊕, ⊗ : F 2 → F be definable functions such that (F, <, ⊕, ⊗) is an ordered field isomorphic to (R, <, +, ·). Then there is a definable set Z ⊆ F such that (F, <, ⊕, ⊗, Z) is isomorphic to (R, <, +, ·, Z). Proof: Let ι : (F, <, ⊕, ⊗) → (R, <, +, ·) be an isomorphism. Enough to show that ι−1(Z) is definable, because then ι : (F, <, ⊕, ⊗, Z) → (R, <, +, ·, Z). Let U be the complement of the closure of F. Let D ⊆ F be the definable set of endpoints of bounded connected components of U that lie in F. D is dense in F. Let δ : D → R map d to the length of the connected component of U

• ne of whose endpoints d is.

Set d ≺ d′ if either δ(d′) < δ(d) or (δ(d′) = δ(d) and d < d′).

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 11 / 14

Let F ⊆ R be nowhere dense and ⊕, ⊗ : F 2 → F be definable functions such that (F, <, ⊕, ⊗) is an ordered field isomorphic to (R, <, +, ·). Then there is a definable set Z ⊆ F such that (F, <, ⊕, ⊗, Z) is isomorphic to (R, <, +, ·, Z). Proof: Let ι : (F, <, ⊕, ⊗) → (R, <, +, ·) be an isomorphism. Enough to show that ι−1(Z) is definable, because then ι : (F, <, ⊕, ⊗, Z) → (R, <, +, ·, Z). Let U be the complement of the closure of F. Let D ⊆ F be the definable set of endpoints of bounded connected components of U that lie in F. D is dense in F. Let δ : D → R map d to the length of the connected component of U

• ne of whose endpoints d is.

Set d ≺ d′ if either δ(d′) < δ(d) or (δ(d′) = δ(d) and d < d′). It is easy to see that ≺ is an ω-order on D. Conclude that (F, <, ⊕, ⊗, D, ≺) defines ι−1(Z).

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 11 / 14

An application to metric dimensions

H.-Miller 15. Let R be an expansion of (R, <, +, ·) that is not type C, then the Assouad dimension of any compact definable subset of Rn agrees with its topological dimension.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 12 / 14

An application to metric dimensions

H.-Miller 15. Let R be an expansion of (R, <, +, ·) that is not type C, then the Assouad dimension of any compact definable subset of Rn agrees with its topological dimension.

• Observation. All commonly encountered notions of metric dimension are

bounded from below by the topological dimension and bounded from above the Assouad dimension. No fractals!

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 12 / 14

An application to metric dimensions

H.-Miller 15. Let R be an expansion of (R, <, +, ·) that is not type C, then the Assouad dimension of any compact definable subset of Rn agrees with its topological dimension.

• Observation. All commonly encountered notions of metric dimension are

bounded from below by the topological dimension and bounded from above the Assouad dimension. No fractals!

• Observation. Fails without multiplication. For example, (R, <, +, C) is

type B, where C is the middle-thirds Cantor set.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 12 / 14

An application to metric dimensions

H.-Miller 15. Let R be an expansion of (R, <, +, ·) that is not type C, then the Assouad dimension of any compact definable subset of Rn agrees with its topological dimension.

• Observation. All commonly encountered notions of metric dimension are

bounded from below by the topological dimension and bounded from above the Assouad dimension. No fractals!

• Observation. Fails without multiplication. For example, (R, <, +, C) is

type B, where C is the middle-thirds Cantor set.

• Theorem. The Assouad dimension of any compact definable subset of Rn

agrees with its topological dimension if one of the following holds:

1 R defines a non-affine C 1 function and does not interpret

(R, <, +, ·, Z).

2 R defines a non-affine C 2 function and is not type C. Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 12 / 14

• Definition. A decreasing sequence with decreasing gaps is a strictly

decreasing sequence (sn)n∈N with limit zero such that (sn+1 − sn)n∈N is also strictly decreasing. We say that (sn)n∈N has exponential decay if for some λ < 0 we have sn < exp(λn) for sufficiently large n. Otherwise, we say the sequence has subexponential decay.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 13 / 14

• Definition. A decreasing sequence with decreasing gaps is a strictly

decreasing sequence (sn)n∈N with limit zero such that (sn+1 − sn)n∈N is also strictly decreasing. We say that (sn)n∈N has exponential decay if for some λ < 0 we have sn < exp(λn) for sufficiently large n. Otherwise, we say the sequence has subexponential decay. Fix a decreasing sequence (sn)n∈N with decreasing gaps and set S = {sn : n ∈ N}.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 13 / 14

• Definition. A decreasing sequence with decreasing gaps is a strictly

decreasing sequence (sn)n∈N with limit zero such that (sn+1 − sn)n∈N is also strictly decreasing. We say that (sn)n∈N has exponential decay if for some λ < 0 we have sn < exp(λn) for sufficiently large n. Otherwise, we say the sequence has subexponential decay. Fix a decreasing sequence (sn)n∈N with decreasing gaps and set S = {sn : n ∈ N}. Garcia, Hare and Mendiv (2016). The following are equivalent: S has positive Assouad dimension, S has Assouad dimension one, (sn)n∈N has subexponential decay.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 13 / 14

• Definition. A decreasing sequence with decreasing gaps is a strictly

decreasing sequence (sn)n∈N with limit zero such that (sn+1 − sn)n∈N is also strictly decreasing. We say that (sn)n∈N has exponential decay if for some λ < 0 we have sn < exp(λn) for sufficiently large n. Otherwise, we say the sequence has subexponential decay. Fix a decreasing sequence (sn)n∈N with decreasing gaps and set S = {sn : n ∈ N}. Garcia, Hare and Mendiv (2016). The following are equivalent: S has positive Assouad dimension, S has Assouad dimension one, (sn)n∈N has subexponential decay.

• Corollary. Suppose that (sn)n∈N has subexponential decay. Let

f : [0, 1] → R be a non-affine function. If f is C 1, then (R, <, +, S, f ) interprets (R, <, +, ·, Z). If f is C 2, then (R, <, +, S, f ) is type C.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 13 / 14

• Corollary. Suppose that (sn)n∈N has subexponential decay. Let

f : [0, 1] → R be a non-affine function. If f is C 1, then (R, <, +, S, f ) interprets (R, <, +, ·, Z). If f is C 2, then (R, <, +, S, f ) is type C. Optimality I. By Friedman-Miller any expansion of (R, <, +, (x → λx)λ∈R) by a countable compact subset of R is type A.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 14 / 14

• Corollary. Suppose that (sn)n∈N has subexponential decay. Let

f : [0, 1] → R be a non-affine function. If f is C 1, then (R, <, +, S, f ) interprets (R, <, +, ·, Z). If f is C 2, then (R, <, +, S, f ) is type C. Optimality I. By Friedman-Miller any expansion of (R, <, +, (x → λx)λ∈R) by a countable compact subset of R is type A. Optimality II. By van den Dries the expansion of (R, <, +, ·) by {2−n : n ∈ N} is type A.

Philipp Hieronymi (Illinois) On continuous functions Bedlewo July 2017 14 / 14