aa r X i v : . [ m a t h . L O ] A p r DP-MINIMAL EXPANSIONS OF ( Z , + ) VIA DENSE PAIRS VIAMORDELL-LANG
ERIK WALSBERG
Abstract.
This is a contribution to the classification problem for dp-minimalexpansions of ( Z , + ) . Let S be a dense cyclic group order on ( Z , + ) . We use re-sults on “dense pairs” to construct uncountably many dp-minimal expansionsof ( Z , + , S ) . These constructions are applications of the Mordell-Lang con-jecture and are the first examples of “non-modular” dp-minimal expansionsof ( Z , + ) . We canonically associate an o-minimal expansion R of ( R , + , × ) ,an R -definable circle group H , and a character Z → H to a “non-modular”dp-minimal expansion of ( Z , + , S ) . We also construct a “non-modular” dp-minimal expansion of ( Z , + , Val p ) from the character Z → Z × p , k ↦ Exp ( pk ) . Introduction
We construct new dp-minimal expansions of ( Z , + ) and take some steps towardsclassifying dp-minimal expansions of ( Z , + ) which define either a dense cyclic grouporder or a p -adic valuation. (Every known proper dp-minimal expansion of ( Z , + ) defines either a dense cyclic group order, a p -adic valuation, or < .)We recall the definition of dp-minimality in Section 3. Dp-minimality is a strongform of NIP which is broad enough to include many interesting structures and nar-row enough to have very strong consequences. O-minimality and related notionsimply dp-minimality. Johnson [20] classified dp-minimal fields. Simon [40] showedthat an expansion of ( R , + , < ) is dp-minimal if and only if it is o-minimal. Wesummarize recent work on dp-minimal expansions of ( Z , + ) in Section 6.It was an open question for some years whether every proper dp-minimal expansionof ( Z , + ) is interdefinable with ( Z , + , < ) [3, Question 5.32]. It turns out that thisquestion was essentially answered before it was posed, in work on “dense pairs”.We will show, applying work of Hieronymi and G¨unaydin [16], that if S is the unitcircle, t ∈ R is irrational, and χ ∶ Z → S is the character χ ( k ) ∶ = e πitk then thestructure induced on Z by ( R , + , × ) and χ is dp-minimal.Indeed, for every known dp-minimal expansion Z of ( Z , + ) there is a dp-minimalfield K , a semiabelian K -variety V , and a character χ ∶ Z → V ( K ) such that thestructure Z χ induced on Z by K and χ is dp-minimal and Z is a reduct of Z χ .We now briefly describe how the known dp-minimal expansions of ( Z , + ) fall intothis framework. It follows directly from the Mordell-Lang conjecture that if β ∈ C × is not a root of unity then the structure induced on Z by ( C , + , × ) and the character Date : April 16, 2020. k ↦ β k is interdefinable with ( Z , + ) . It follows from a result of Tychonievich [46,Theorem 4.1.2] that if β ∈ R × \{ − , } then the structure induced on Z by ( R , + , × ) and the character k ↦ β k is interdefinable with ( Z , + , < ) . (It is also shown in [29]that if β ∈ Q × p and Val p ( β ) ≠ ( Z , + , < ) is interdefinable with the structureinduced on Z by ( Q p , + , × ) and k ↦ β k .) Below we apply work of Mariaule [29] toshow that there is β ∈ + p Z p such that the structure induced on Z by ( Q p , + , × ) and k ↦ β k is a dp-minimal expansion of ( Z , + , Val p ) . The only other previouslyknown dp-minimal expansion of ( Z , + ) is ( Z , + , S ) where S is a dense cyclic grouporder [45]. There is a unique β ∈ S such that S is the pullback of the clockwisecyclic order on S by k ↦ β k . So the structure induced on Z by ( R , + , × ) and k ↦ β k is a dp-minimal expansion of ( Z , + , S ) .We produce uncountably many new dp-minimal expansions of ( Z , + , S ) . Let E bean elliptic curve defined over R , E ( R ) be the connected component of the identity,and χ ∶ Z → E ( R ) be a character such that S is the pullback by χ of the naturalcyclic order on E ( R ) . We apply [16] to show that the structure Z E induced on Z by ( R , + , × ) and χ is a proper dp-minimal expansion of ( Z , + , S ) . We also showthat E ( R ) may be recovered up to semialgebraic isomorphism from Z E . It followsthat there is an uncountable family of dp-minimal expansions of ( Z , + , S ) no twoof which are interdefinable.We describe how E ( R ) may be recovered from Z E . Let C be the usual clockwisecyclic order on R / Z . Given any dp-minimal expansion Z of ( Z , + , S ) we define acompletion Z □ of Z , this Z □ is an o-minimal expansion of ( R / Z , + , C ) canonicallyassociated to Z . We show that Z □ E is the structure induced on R / Z by ( R , + , × ) and the unique (up to sign) topological group isomorphism R / Z → E ( R ) . Therecovery of E ( R ) from Z □ E is a special case of a canonical correspondence between(1) non-modular o-minimal expansions C of ( R / Z , + , C ) , and(2) pairs ⟨ R , H ⟩ where R is an o-minimal expansion of R and H is an R -definablecircle group.Given ⟨ R , H ⟩ , C is unique up to interdefinibility. Given C , R is unique up to inter-definibility and H is unique up to R -definable isomorphism.We describe Z □ for a fixed dp-minimal expansion Z of ( Z , + , S ) . Let ψ ∶ Z → R / Z be the unique character such that S is the pullback of C by ψ . Let Z ≺ N be highlysaturated, N Sh be the Shelah expansion of N , and Inf be the natural subgroup ofinfinitesimals in N . We identify N / Inf with R / Z and identify the quotient map N → R / Z with the standard part map. As Inf is N Sh -definable we regard R / Z as an imaginary sort of N Sh . A slight adaptation of [49] shows that the followingstructures are interdefinable:(1) The structure on R / Z with an n -ary relation defining the closure in ( R / Z ) n of {( ψ ( a ) , . . . , ψ ( a n )) ∶ ( a , . . . , a n ) ∈ X } for each Z Sh -definable X ⊆ Z n ,(2) The structure on R / Z with an n -ary relation defining the image under thestandard part map N n → ( R / Z ) n of each N -definable subset of N n ,(3) The structure induced on R / Z by N Sh . We refer to any of these structure as Z □ . It follows from ( ) that Z □ is dp-minimal,a slight adaptation of [40] shows that any dp-minimal expansion of ( R / Z , + , C ) iso-minimal, so Z □ is o-minimal. We will see that the structure induced on Z by Z □ and ψ is a reduct of the Shelah expansion of Z . In future work we intend toshow that these two are interdefinable. This will reduce the question “what arethe dp-minimal expansions of ( Z , + , S ) to “for which o-minimal expansions C of ( R / Z , + , C ) is the structure induced on Z by C and ψ dp-minimal”?We also define an analogous completion P □ of a dp-minimal expansion P of ( Z , + , Val p ) ,this P □ is a dp-minimal expansion of ( Z p , + , Val p ) . The structure induced on Z by P □ is reduct of P Sh . We expect the induced structure to be interdefinable with P Sh .It is easy to see that Z □ E defines an isomorphic copy of ( R , + , × ) . It follows that if Z E ≺ N E is highly saturated then the Shelah expansion of N E interprets ( R , + , × ) ,so Z E should be “non-modular”. (One can show that Z E itself does not interpretan infinite field.) At present there is no published notion of modularity for generalNIP structures, but there should be a notion of modularity for NIP (or possiblyjust distal) structures which satisfies the following.(A1) A modular structure cannot interpret an infinite field.(A2) Abelian groups, linearly (or cyclically) ordered abelian groups, NIP valuedabelian groups, and ordered vector spaces are modular.(A3) If M is modular and the structure induced on A ⊆ M n by M eliminatesquantifiers then the induced structure is modular. In particular the Shelahexpansion of a modular structure is modular. (Recall that the inducedstructure eliminates quantifiers if and only if every definable subset of A m is of the form A m ∩ X for M -definable X .)(A4) An o-minimal structure is modular if and only if it does not define aninfinite field. (This should follow from the Peterzil-Starchenko trichotomy.)In this paper we will assume that there is a notion of modularity satisfying theseconditions, but none of our results fail if this is not true. ( A2 ) implies that allpreviously known dp-minimal expansions of ( Z , + ) are modular. ( A1 ) and ( A3 ) imply that if Z □ defines ( R , + , × ) then Z is non-modular. If Z □ does not define ( R , + , × ) then ( A4 ) implies that Z □ is modular. We expect that if Z □ is modularthen Z is modular.We will see that if P is the structure induced on Z by ( Q p , + , × ) and the charac-ter k ↦ Exp ( pk ) then P □ is interdefinable with the structure induced on Z p by ( Q p , + , × ) and the isomorphism ( Z p , + ) → ( + p Z p , × ) , a ↦ Exp ( pa ) . It followsthat the Shelah expansion of a highly saturated P ≺ N interprets ( Q p , + , × ) , so P is non-modular. We again expect that P is modular if and only if P □ is modular,but we do not have a modular/non-modular dichotomy for dp-minimal expansionsof ( Z p , + , Val p ) (we lack a p -adic Peterzil-Starchenko.) It seems reasonable to con-jecture that a dp-minimal expansion of ( Z p , + , Val p ) is non-modular if and only ifit defines an isomorphic copy of ( Q p , + , × ) .We now summarize the sections. In Section 3 we recall some background model-theoretic notions, in Section 4 we recall background on cyclically ordered abelian ERIK WALSBERG groups, and in Section 5 we recall some basic facts on definable groups in o-minimalexpansions of ( R , + , × ) . In Section 6 we survey previous work on dp-minimal expan-sions of ( Z , + ) . In Section 7 we construct new dp-minimal expansions of ( Z , + , S ) where S is a dense cyclic group order. In Section 8 we describe the o-minimalcompletion of a strongly dependent expansion of ( Z , + , S ) . We also show that theShelah expansion ( Z , + , S ) Sh of ( Z , + , S ) is interdefinable with the structure in-duced on Z by ( R / Z , + , C ) and ψ , where ψ ∶ Z → R / Z is the unique charactersuch that S is the pullback of C by ψ . It follows that ( Z , + , S ) Sh is a reduct ofeach of our dp-minimal expansions of ( Z , + , S ) . In Section 9 we show that two ofour dp-minimal expansions of ( Z , + , S ) are interdefinable if and only if the asso-ciated semialgebraic circle groups are semialgebraically isomorphic. In Section 10we construct a new dp-minimal expansion P of ( Z , + , Val p ) and in Section 11 wedescribe the p -adic completion of a dp-minimal expansion of ( Z , + , Val p ) . In Sec-tion 12 we give a conjecture which implies that one can construct uncountablymany dp-minimal expansions of ( Z , + , Val p ) from p -adic elliptic curves. Finally, inSection 13 we briefly discuss the question of whether our completion constructionsare special cases of an abstract model-theoretic completion.1.1. Acknowledgements.
Thanks to Philipp Hieronymi for various discussions ondense pairs and thanks to the audience of the Berkeley logic seminar for showinginterest in a talk that turned into this paper. This paper owes a profound debtto Minh Chieu Tran. He proposed that if E is an elliptic curve defined over Q and the group E ( Q ) of Q -points of E is isomorphic to ( Z , + ) , then the structureinduced on E ( Q ) by ( Q p , + , × ) might be dp-minimal and that one might therebyproduce a new dp-minimal expansion of ( Z , + ) . Conjecture 4 is a modification ofthis idea (there does not appear to be anything to be gained by restricting to E ( Q ) as opposed to other infinite cyclic subgroups of E ( Q p ) , or by assuming that E isdefined over Q as opposed to Q p .)2. Conventions, notation, and terminology
Given a tuple x = ( x , . . . , x n ) of variables we let ∣ x ∣ = n . Throughout n is anatural number, m, k, l are integers, t, r, λ, η are real numbers, and α is an elementof R / Z . Suppose α ∈ R / Z . We let ψ α denote the character Z → R / Z given by ψ α ( k ) = αk . We say that α is irrational if α = s + Z for s ∈ R \ Q . Note that α is irrational if and only if ψ α is injective.All structures are first order and “definable” means “first-order definable, possiblywith parameters”. Suppose M , N , and O are structures on a common domain M .Then M is a reduct of O (and O is an expansion of M ) if every M -definable subsetof every M n is O -definable, M and O are interdefinable if each is a reduct of theother, M is a proper reduct of O (and O is a proper expansion of M ) if M is areduct of O and M is not interdefinable with O , and N is intermediate between M and O if M is a proper reduct of N and N is a proper reduct of O .Given a set A and an injection f ∶ A → M m we say that the structure in-duced on A by M and f is the structure on A with an n -ary relation defining {( a , . . . , a n ) ∈ A n ∶ (( f ( a ) . . . , f ( a n )) ∈ Y } for every M -definable Y ⊆ M nm .If A is a subset of M m and f ∶ A → M m is the identity we refer to this as the structure induced on A by M .We let Cl ( X ) denote the closure of a subset X of a topological space.Suppose L ⊆ L ′ are languages containing < , R ′ is an L ′ -structure expanding ( R , < ) and R is the L -reduct of R ′ . The open core of R ′ is the reduct of R ′ generatedby all closed R ′ -definable sets. Furthermore Th ( R ) is an open core of Th ( R ′ ) if,whenever R ′ ≺ N ′ then the L ′ -reduct of N ′ is interdefinable with the open core of N ′ . This notion clearly makes sense in much broader generality.We use “semialgebraic” as a synonym of either “ ( R , + , × ) -definable” or “ ( Q p , + , × ) -definable”. It will be clear in context which we mean.3. Model-theoretic preliminaries
Let M be a structure and M ≺ N be highly saturated.3.1. Dp-minimality.
Our reference is [41]. Recall that M is dp-minimal if forevery small set A of parameters from N , pair I , I of mutually indiscernible se-quences in N over A , and b ∈ N , I i is indiscernible over A ∪ { b } for some i ∈ { , } .We now describe a second definition of dp-minimality which will be useful below.A family ( θ i ∶ i ∈ I ) of formulas is n -inconsistent if ⋀ i ∈ J θ i is inconsistent for every J ⊆ I, ∣ J ∣ = n . A pair ϕ ( x ; y ) , φ ( x ; z ) of formulas and n ∈ N violate inp-minimality if ∣ x ∣ = k ≥ a , . . . , a k ∈ M ∣ y ∣ and b , . . . , b k ∈ M ∣ z ∣ such that ϕ ( x ; a ) , . . . , ϕ ( x ; a k ) and φ ( x ; b ) , . . . , φ ( x ; b k ) are both n -inconsistentand M ⊧ ∃ x [ ϕ ( x ; a i ) ∧ φ ( x ; b j )] for any 1 ≤ i, j ≤ k . We say that ϕ ( x ; y ) and φ ( x ; z ) violate inp-minimality if there is n such that ϕ ( x ; y ) , φ ( x ; z ) , n violate inp-minimality. Then M is inp-minimal if no pair of formulas violates inp-minimality.Recall that M is dp-minimal if and only if M is inp-minimal and NIP.Fact 3.1 is an easy application of Ramsey’s theorem which we leave to the reader. Fact 3.1.
Let ϕ ( x ; y ) , . . . , ϕ m ( x ; y m ) and φ ( x ; z ) , . . . , φ m ( x ; z m ) be formulas.If ϕ ∪ ( x ; y , . . . , y m ) = m ⋁ i = ϕ i ( x ; y i ) , φ ∪ ( x ; z , . . . , z m ) ∶ = m ⋁ i = φ i ( x ; z i ) violate inp-minimality then ϕ i ( x ; y i ) , φ j ( x ; z j ) violate inp-minimality for some i, j . We also leave the proof of Fact 3.2 to the reader.
Fact 3.2.
Fix formulas ϕ ( x ; y ) , φ ( x ; y ) with ∣ x ∣ = . Suppose there is n such that M ⊧ ∀ y ∃ ≤ n xϕ ( x ; y ) . Then ϕ ( x ; y ) and φ ( x ; y ) do not violate inp-minimality. External definibility.
A subset of X of M n is externally definable ifthere is an N -definable subset Y of N n such that X = M n ∩ Y . By saturation thecollection of externally definable sets does not depend on choice of N . The Shelahexpansion M Sh of M is the expansion by all externally definable subsets of all M n ,equivalently, the structure induced on M by N . We will make frequent use of thefollowing elementary observation. ERIK WALSBERG
Fact 3.3.
Suppose that M expands a linear order. Then every convex subset of M is externally definable. The first claim of Fact 3.4 is a theorem of Shelah [38], see also Chernikov andSimon [7]. The latter claims follow easily from the first, see for example Onshuusand Usvyatsov [33].
Fact 3.4. If M is NIP then every M Sh -definable subset of every M n is externallydefinable in M . If M is NIP then M Sh is NIP , if M is strongly dependent then M Sh is strongly dependent, and if M is dp-minimal then M Sh is dp-minimal. Fact 3.5 is a theorem of Chernikov and Simon [8, Corollary 9].
Fact 3.5.
Suppose M is NIP and X is an externally definable subset of M n . Thenthere is an M -definable family ( X a ∶ a ∈ M m ) of subsets of M n such that for everyfinite B ⊆ X there is a ∈ M m such that B ⊆ X a ⊆ X . Fact 3.6 is the Marker-Steinhorn theorem [30].
Fact 3.6.
Suppose R is an o-minimal expansion of ( R , < ) . Every externally defin-able subset of every R n is definable. Equivalently: R Sh and R are interdefinable. Fact 3.7 is a theorem of Delon [11].
Fact 3.7.
Every subset of Q np which is externally definable in ( Q p , + , × ) is definablein ( Q p , + , × ) . Equivalently: ( Q p , + , × ) Sh and ( Q p , + , × ) are interdefinable. Weak minimality.
Suppose O expands M . We say that O is M -minimal if every O -definable subset of M is definable in M and we say that O is weakly M -minimal if every O -definable subset of M is externally definable in M .Suppose L ⊆ L ′ are languages, T ′ is a complete consistent L ′ -theory, and T is the L -reduct of T ′ . We say that T ′ is T - minimal if for every L ′ -formula ϕ ( x ; y ) , ∣ x ∣ = L -formula φ ( x ; z ) such that for every P ⊧ T ′ and a ∈ P ∣ y ∣ there is b ∈ P ∣ z ∣ such that ϕ ( P ; a ) = φ ( P ; b ) . We say that T ′ is weakly T - minimal iffor every L ′ -formula ϕ ( x ; y ) , ∣ x ∣ = L -formula φ ( x ; z ) such that forevery P ⊧ T ′ , highly saturated P ≺ Q , and a ∈ P ∣ y ∣ , there is b ∈ Q ∣ z ∣ such that ϕ ( P ; a ) = P ∩ φ ( Q ; b ) . A structure is weakly T -minimal if its theory is.Weak minimality was introduced in [42]. If T is a complete theory of dense lin-ear orders then T ′ is T -minimal if and only if T ′ is o-minimal and T ′ is weakly T -minimal if and only if T ′ is weakly o-minimal.Suppose ⭑ is an NIP-theoretic property such that T has ⭑ if and only if every T -model omits a certain configuration involving only unary definable sets. It isthen easy to see that if T is ⭑ and T ′ is weakly T -minimal then T ′ is ⭑ . Fact 3.8.
Suppose T ′ is weakly T -minimal. If T satisfies any one of the followingproperties, then so does T ′ .(1) stability,(2) NIP ,(3) strong dependence,(4) dp-minimality. Cyclically ordered abelian groups
We give basic definitions and results concerning cyclically ordered groups. We alsoset notation to be used throughout. See [45] for more information and references.A cyclic order S on a set G is a ternary relation such that for all a, b, c ∈ G ,(1) if S ( a, b, c ) , then S ( b, c, a ) ,(2) if S ( a, b, c ) , then ¬ S ( c, b, a ) ,(3) if S ( a, b, c ) and S ( a, c, d ) then S ( a, b, d ) ,(4) if a, b, c are distinct, then either S ( a, b, c ) or S ( c, b, a ) .An open S -interval is a set of the form { b ∈ G ∶ S ( a, b, c )} for some a, c ∈ G ,likewise define closed and half open intervals. A subset of G is S -convex if it is theunion of a nested family of intervals. We drop the “ S ” when it is clear from context.If ( G, + ) is an abelian group then a cyclic group order on ( G, + ) is a + -invariantcyclic order. Suppose S is a cyclic group order on ( G, + ) . A subset of G is an Stmc set if it is of the form a + mJ for S -convex J ⊆ G and a ∈ G . We drop the“ S ” when it is clear from context.Note that {( a, b, c ) ∈ G ∶ S ( c, b, a )} is a cyclic group order which we refer to as the opposite of S . (If < is a linear group order on ( G, + ) then {( a, b ) ∈ G ∶ b < a } isalso a linear group order which we refer to as the opposite of < .)Throughout C is the cyclic group order on ( R / Z , + ) such that whenever t, t ′ , t ′′ ∈ R and 0 ≤ t, t ′ , t ′′ < C ( t + Z , t ′ + Z , t ′′ + Z ) holds if and only if either t < t ′ < t ′′ , t ′ < t ′′ < t , or t ′′ < t < t ′ . Given irrational α ∈ R / Z we let C α C α C α be the cyclicgroup order on ( Z , + ) where C α ( k, k ′ , k ′′ ) if and only if C ( αk, αk ′ , αk ′′ ) , so C α isthe pullback of C by ψ α . Every dense cyclic group order on ( Z , + ) is of this formfor unique α ∈ R / Z .Let < be a linear group order on ( G, + ) . There are two associated cyclic orders: S < ∶ = {( a, b, c ) ∈ G ∶ ( a < b < c ) ∨ ( b < c < a ) ∨ ( c < a < b )} , and S > ∶ = {( a, b, c ) ∈ G ∶ ( c < b < a ) ∨ ( b < a < c ) ∨ ( a < c < b )} . Note that S < is the opposite of S > . See for example [45] for a proof of Fact 4.1. Fact 4.1.
Every cyclic group order on ( Z , + ) is either C α for some irrational α ∈ R / Z or S < or S > for the usual order < . We will frequently apply Fact 4.2, which is elementary and left to the reader.
Fact 4.2.
Suppose H is a topological group and γ is an isomorphism H → R / Z oftopological groups. Then γ is unique up to sign, i.e. if ξ ∶ H → R / Z is a topologicalgroup isomorphism then either ξ = γ or ξ = − γ . The universal cover.
We describe the universal cover of ( G, + , S ) . A uni-versal cover of ( G, + , S ) is an ordered abelian group ( H, + , < ) , a distinguishedpositive u ∈ H such that u Z is cofinal in H , and a surjective group homomor-phism π ∶ H → G with kernel u Z such that if a, b, c ∈ H and 0 ≤ a, b, c < u then S ( π ( a ) , π ( b ) , π ( c )) if and only if we either have a < b < c , b < c < a , or c < a < b . ERIK WALSBERG
The universal cover ( H, + , < , u, π ) is unique up to unique isomorphism and everycyclically ordered abelian group has a universal cover.So ( R , + , < , , π ) is a universal cover of ( R / Z , + , C ) , where π ( t ) = t + Z for all t ∈ R and ( Z + s Z , + , < , , π ) is a universal cover of ( Z , + , C α ) when α = s + Z .5. Definable groups
We recall some basic facts from the extensive theory of definable groups in o-minimal structures.
Throughout this section R is an o-minimal expansionof ( R , + , × ) , H is an R -definable group, and “dimension” without modifi-cation is the o-minimal dimension. Fact 5.1 follows from work of Pillay [36] and [47, 10.1.8].
Fact 5.1.
There is an R -definable group G with underlying set G ⊆ R m such that G is a topological group with respect to the topology induced by R m and an R -definablegroup isomorphism ξ ∶ H → G . If G ′ is an R -definable group with underlying set G ′ ⊆ R n , G ′ is a topological group with respect to the topology induced by R n , and ξ ′ ∶ H → G ′ is an R -definable group isomorphism, then ξ ′ ◦ ξ − is a topological groupisomorphism G → G ′ . We let T H be the canonical group topology on H and consider H as a topologicalgroup. Recall that any connected topological group of topological dimension one isisomorphic (as a topological group) to either ( R , + ) or ( R / Z , + ) . It follows that if H is one-dimensional and connected then H is isomorphic as a topological group toeither ( R , + ) or ( R / Z , + ) . In the first case we say that H is a line group , in thesecond case H is a circle group .Suppose X is an R -definable subset of R m . An easy application of the good direc-tions lemma [47, Theorem 4.2] shows that if X is homeomorphic to R then thereis an R -definable homeomorphism X → R and if X is homeomorphic to R / Z thenthere is an R -definable homeomorphism from X to the unit circle. (The analogousfact fails in higher dimensions, there are homeomorphic semialgebraic sets X, X ′ for which there is no homeomorphism X → X ′ definable in an o-minimal expan-sion of ( R , + , × ) , this is a consequence of Shiota’s o-minimal Hauptvermutung [39]together with the failure of the classical Hauptvermutung.) Fact 5.2 easily follows. Fact 5.2.
Suppose H is one-dimensional, connected, and has underlying set H andgroup operation ⊕ . Then there is a unique up to opposite R -definable cyclic grouporder S on H . If H is a line group then ( H, + , S ) is isomorphic to ( R , + , S < ) . If H is a circle group the ( H, + , S ) is isomorphic to ( R / Z , + , C ) . So if H is one-dimensional and connected and A is a subgroup of H then we mayspeak without ambiguity of a tmc subset of A .Finally we recall the interpretation-rigidity theorem for o-minimal expansions of ( R , + , × ) . Fact 5.3 is due to Otero, Peterzil, and Pillay [34]. Fact 5.3.
Let F be an infinite field interpretable in R . Then there is either an R -definable field isomorphism F → ( R , + , × ) or F → ( C , + , × ) . It follows that if anexpansion S of ( R , + , × ) is interpretable in R then S is isomorphic to a reduct of R , and if a structure M is mutually interpretable with R then R is (up to interdefini-bility) the unique expansion of ( R , + , × ) mutually interpretable with M . What we know about dp-minimal expansions of ( Z , + ) We survey what is known about dp-minimal expansions of ( Z , + ) .The first result on dp-minimal expansions of ( Z , + ) is Fact 6.1, proven in [4, Propo-sition 6.6]. Fact 6.1 follows easily from two results, the Michaux-Villemaire theo-rem [32] that there are no proper ( Z , + , < ) -minimal expansions of ( Z , + , < ) , andSimon’s theorem [40, Lemma 2.9] that a definable family of unary sets in a dp-minimal expansion of a linear order has only finitely many germs at infinity. Fact 6.1.
There are no proper dp-minimal expansions of ( Z , + , < ) . Equivalently:there are no proper dp-minimal expansions of ( N , + ) . The authors of [4] raised the question of whether there is a dp-minimal expansionof ( Z , + ) which is not a reduct of ( Z , + , < ) . Conant and Pillay [10] proved Fact 6.2.Their proof relies on earlier work of Palac´ın and Sklinos [35], who apply the Buechlerdichotomy theorem and other sophisticated tools of stability theory. Fact 6.2.
There are no proper stable dp-minimal expansions of ( Z , + ) . Conant [9] proved Fact 6.3 via a geometric analysis of ( Z , + , < ) -definable sets.Facts 6.2 and 3.8 show that a proper dp-minimal expansion of ( Z , + ) is not Th ( Z , + ) -minimal. Alouf and d’Elb´ee [2] used this to give a quicker proof of Fact 6.3. Fact 6.3.
There are no intermediate structures between ( Z , + ) and ( Z , + , < ) . Alouf and d’Elb´ee [2] proved Fact 6.4. Given a prime p we let Val p be the p -adic valuation on ( Z , + ) and ≺ p be the partial order on Z where m ≺ p n if andonly if Val p ( m ) < Val p ( n ) . We can view ( Z , + , Val p ) as either ( Z , + , ≺ p ) or asthe two sorted structure with disjoint sorts Z and N ∪ { ∞ } , addition on Z , andVal p ∶ Z → N ∪ { ∞ } . It makes no difference which of these two options we take. Fact 6.4.
Let p be a prime. Then ( Z , + , Val p ) is dp-minimal and ( Z , + ) -minimal,and there are no structures intermediate between ( Z , + ) and ( Z , + , Val p ) . Alouf and d’Elb´ee also show that ( Z , + , ( Val p ) p ∈ I ) has dp-rank ∣ I ∣ for any nonemptyset I of primes. So if p ≠ q are primes then ( Z , + , Val p ) and ( Z , + , Val q ) do nothave a common dp-minimal expansion.So far we have described countably many dp-minimal expansions of ( Z , + ) . Fact 6.5,proven by Tran and Walsberg [45], shows that there is an uncountable collection ofdp-minimal expansions of ( Z , + ) , no two of which are interdefinable. Fact 6.5.
Suppose α, β ∈ R / Z are irrational. Then ( Z , + , C α ) is dp-minimal.Furthermore ( Z , + , C α ) and ( Z , + , C β ) are interdefinable if and only if α and β are Z -linearly dependent. Fact 6.5, Fact 4.1, and dp-minimality of ( Z , + , < ) together show that any expansionof ( Z , + ) by a cyclic group order is dp-minimal.It is shown in [45] that every unary definable set in every elementary extension of ( Z , + , C α ) is a finite union of tmc sets. It follows by Fact 3.8 that if Z expands ( Z , + , C α ) and every unary definable set in every elementary extension of Z is afinite union of tmc sets, then Z is dp-minimal. A converse is proven in [42]. Fact 6.6.
Fix irrational α ∈ R / Z . Suppose Z is a dp-minimal expansion of ( Z , + , C α ) . Then Z is weakly Th ( Z , + , C α ) -minimal (equivalently: every unarydefinable set in every elementary extension of Z is a finite union of tmc sets). In particular a dp-minimal expansion of ( Z , + , C α ) Sh cannot add new unary sets.Suppose α, β ∈ R / Z are irrational and Z -linearly independent. An easy applicationof Kronecker density shows that if I is an infinite and co-infinite C α -convex set then I is not a finite union of C β -tmc sets, see [45]. Fact 6.7 follows. Fact 6.7.
Suppose α, β ∈ R / Z are irrational and Z -linearly independent. Suppose Z α is a dp-minimal expansion of ( Z , + , C α ) and Z β is a dp-minimal expansion of ( Z , + , C β ) . If I is an infinite and co-infinite C α -interval then I is not Z β -definable,and vice versa. So Z α defines a subset of Z which is not Z β -definable, and viceversa. In particular Z α and Z β do not have a common dp-minimal expansion. We now describe a striking recent result of Alouf [1]. We first recall Fact 6.8, aspecial case of [19, Lemma 3.1].
Fact 6.8.
Suppose G is a dp-minimal expansion of a group G which defines a non-discrete Hausdorff group topology on G . Then G eliminates ∃ ∞ . Fact 6.8 shows that any dp-minimal expansion of ( Z , + , Val p ) or ( Z , + , C α ) elimi-nates ∃ ∞ . Fact 6.9 is proven in [1]. Fact 6.9.
Suppose Z is a dp-minimal expansion of ( Z , + ) which either(1) does not eliminate ∃ ∞ ,(2) or defines an infinite subset of N .Then Z defines < . So ( Z , + , < ) is, up to interdefinibility, the only dp-minimal expansion of ( Z , + ) which does not eliminate ∃ ∞ . Conjecture 1 is now natural. Conjecture 1.
Any proper dp-minimal expansion of ( Z , + ) which eliminates ∃ ∞ defines a non-discrete group topology on ( Z , + ) . Johnson [21] shows that a dp-minimal expansion of a field which is not stronglyminimal admits a definable non-discrete field topology. His proof makes crucial useof the fact that any dp-minimal expansion of a field eliminates ∃ ∞ .6.1. Interpretations.
We describe what we know about interpretations betweendp-minimal expansions of ( Z , + ) . We suspect that bi-interpretable dp-minimalexpansions of ( Z , + ) are interdefinable. Proposition 6.10.
Fix irrational α ∈ R / Z . Suppose Z is a dp-minimal expan-sion of ( Z , + , C α ) . Then Z eq eliminates ∃ ∞ , so Z does not interpret ( Z , + , < ) or ( Z , + , Val p ) for any prime p . Note that ( Z , + , Val p ) eq does not eliminate ∃ ∞ as ( Z , + , Val p ) interprets ( N , < ) .Given a structure M we say that M eq eliminates ∃ ∞ in one variable if for everydefinable family ( E a ∶ a ∈ M k ) of equivalence relations on M there is n such that for all a ∈ M k we either have ∣ M / E a ∣ < n or ∣ M / E a ∣ ≥ ℵ . Proposition 6.10requires Fact 6.11, which is routine and left to the reader. Fact 6.11.
Let M ≺ N be highly saturated. Suppose that N eliminates ∃ ∞ and thereis no N -definable equivalence relation on N with infinitely many infinite classes.Then M eq eliminates ∃ ∞ . We now prove Proposition 6.10. We use the notation and results of [42], so thereader will need to have a copy of that paper at hand.
Proof.
Let ( H, + , < , u, π ) be a universal cover of ( Z , + , C α ) , I ∶ = ( − u, u ) . So let I be the structure induced on I by Z and π . It is shown in [42] that I and Z defineisomorphic copies of each other, so it suffices to show that I eq eliminates ∃ ∞ . Let I ≺ J be highly saturated. The proof of Fact 6.8 shows that J eliminates ∃ ∞ . Weshow that every J -definable equivalence relation on J has only finitely many infiniteclasses and apply Fact 6.11.Suppose E is a J -definable equivalence relation on J with infinitely many infiniteclasses. By [42, Lemma 8.7] there is a finite partition A of J into J -definable setssuch that every E -class is a finite union of sets of the form K ∩ A for convex K and A ∈ A . Fix A ∈ A which intersects infinitely many E -classes. Note that theintersection of each E -class with A is a finite union of convex sets. Let F be theequivalence relation on J where a < b are F -equivalent if and only if there are a ′ < a < b < b ′ such that a ′ , b ′ ∈ A , a ′ and b ′ are E -equivalent, and a ′ , b ′ lie in thesame convex component of E a ′ ∩ A . It is easy to see that every F -class is convexand there are infinitely many F -classes. However, it is shown in the proof of [42,Lemma 8.7] that any definable equivalence relation on J with convex equivalenceclasses has only finitely many infinite classes. (cid:3) Fact 6.12 is proven in [49, Proposition 5.6].
Fact 6.12.
Suppose Z is an NTP expansion of ( Z , < ) and G is an expansion of agroup G which defines a non-discrete Hausdorff group topology on G . Then Z doesnot interpret G . So in particular an NTP expansion of ( Z , + , < ) does not interpret ( Z , + , C α ) for any irrational α ∈ R / Z or ( Z , + , Val p ) for any prime p . In Section 10 we construct a dp-minimal expansion P of ( Z , + , Val p ) which definesaddition on the value set, so in particular P interprets ( Z , + , < ) .7. New dp-minimal expansions of ( Z , + , C α ) We describe new dp-minimal expansions of ( Z , + , C α ) .7.1. Dense pairs.
We first recall Hieronymi and G¨unaydin [16]. Let H be anabelian semialgebraic group with underlying set H ⊆ R m and group operation ⊕ ,and A be a subgroup of H . Then A has the Mordell-Lang property if for every f ∈ R [ x , . . . , x nm ] the set { a ∈ A n ∶ f ( a ) = } is a finite union of sets of the form {( a , . . . , a n ) ∈ A n ∶ k a ⊕ . . . ⊕ k n a n = b } for some k , . . . , k n ∈ Z , b ∈ A. We say that H is a Mordell-Lang group if every finite rank subgroup of H has theMordell-Lang property. Fact 7.1 is essentially in [16], but see the comments below. Fact 7.1.
Suppose H is a one-dimensional connected Mordell-Lang group. Let A be a dense finite rank subgroup of H . Then ( R , + , × , A ) is NIP , Th ( R , + , × ) is anopen core of Th ( R , + , × , A ) , and every subset of A k definable in ( R , + , × , A ) is afinite union of sets of the form b ⊕ n ( X ∩ A k ) for semialgebraic X and b ∈ A k . Note that the last claim of Fact 7.1 shows that structure induced on A by ( R , + , × ) is interdefinable with the structure induced by ( R , + , × , A ) are interdefinable.The reader will not find the exact statement of the last claim of Fact 7.1 in [16].It is incorrectly claimed in [16, Proposition 3.10] that every subset of A k definablein ( R , + , × , A ) is a finite union of sets of the form X ∩ ( b ⊕ nA k ) where X ⊆ H is semialgebraic. This is true when H is a line group, but fails when H is a circlegroup. If H is a circle group and I is an infinite and co-infinite open interval in H then 2 I is not of this form. A slightly corrected version of the proof of [16,Proposition 3.10] yields the last statement of Fact 7.1 .Proposition 7.2 is proven in [42]. Proposition 7.2.
Suppose ( G, + , S ) is a cyclically order abelian group and G ex-pands ( G, + , S ) . Suppose ∣ G / nG ∣ < ℵ for all n . Then G is dp-minimal if and onlyif every unary definable set in every elementary extension of G is a finite union oftmc sets. So G is dp-minimal if and only if Th ( G ) is weakly Th ( G, + , S ) -minimal. Let A be the structure induced on A by ( R , + , × ) . Fact 7.1 shows that every A -definable unary set is a finite union of tmc sets, and that the same claim holds inevery elementary extension of A . Proposition 7.3 follows. Proposition 7.3. If H is a one-dimensional connected Mordell-Lang group and A is a dense finite rank subgroup of H , then the structure induced on A by ( R , + , × ) is dp-minimal. So if H is a Mordell-Lang circle group and χ ∶ Z → H is an injectivecharacter then the structure induced on Z by ( R , + , × ) and χ is dp-minimal. Of course Proposition 7.3 is only relevant because there are semialgebraic Mordell-Lang circle groups by the general Mordell-Lang conjecture. This is a theorem ofFaltings, Vojta, McQuillan and others, see [31] for a survey.
Fact 7.4. If W is a semiabelian variety defined over C , V is a subvariety of W ,and Γ is a finite rank subgroup of W ( C ) , then Γ ∩ V ( C ) is a finite union of cosetsof subgroups of Γ . So ( R > , × ) , the unit circle equipped with complex multiplication,and the real points of an elliptic curve defined over R are all Mordell-Lang groups. Specific examples.
Suppose H is a semialgebraic group equipped with T H .By [18] there is an open neighbourhood U ⊆ H of the identity, an algebraic group W defined over R , a neighbourhood V ⊆ W ( R ) of the identity, and a semialgebraiclocal group isomorphism U → V . We say that H is semiabelian when W is semia-belian. Suppose H is one-dimensional. Then W is one dimensional, so we may take W ( R ) to be either ( R , + ) , ( R × , × ) , the unit circle, or the real points of an ellipticcurve. In the latter three cases H is semiabelian. Thanks to Philipp Hieronymi for discussions on this point. One-dimensional semialgebraic groups were classified up to semialgebraic isomor-phism by Madden and Stanton [27]. There are three families of semiabelian semi-algebraic circle groups.We describe the first family. Given λ > G λ ∶ = ([ , λ ) , ⊗ λ ) where t ⊗ λ t ′ = tt ′ when tt ′ < λ and t ⊗ λ t ′ = tt ′ λ − otherwise. Let λ, η >
1. The unique (up tosign) topological group isomorphism G λ → G η is t ↦ t log λ η . So G λ and G η aresemialgebraically isomorphic if and only if log λ η ∈ Q . Lemma 7.5.
Fix λ > . Suppose A is a finite rank subgroup of G λ . Then ( R , + , × , A ) is NIP , Th ( R , + , × ) is an open core of Th ( R , + , × , A ) , and the struc-ture induced on A by ( R , + , × , A ) is dp-minimal. We let S be the cyclic order on [ , λ ) where S ( t, t ′ , t ′′ ) if and only if either t < t ′ < t ′′ , t ′ < t ′′ < t , or t ′′ < t < t ′ . So S is the unique (up to opposite) semialgebraic cyclicgroup order on G λ . Proof.
Identify G λ with ( R > / λ Z , × ) and let ρ be the quotient map R > → G λ .So ( R > , × , < , λ, ρ ) is a universal cover of ( G λ , S ) . Let H ∶ = ρ − ( A ) . So H isfinite rank and ( H, × , < , λ, ρ ) is a universal cover of ( A, ⊗ λ , S ) . As ( R > , × ) is aMordell-Lang group and H is dense in R > , ( R , + , × , H ) is NIP, Th ( R , + , × ) is anopen core of Th ( R , + , × , H ) , and the structure induced on H by ( R , + , × , H ) isdp-minimal. Observe that A is definable in ( R , + , × , H ) . So ( R , + , × , A ) is NIPand Th ( R , + , × ) is an open core of Th ( R , + , × , A ) . Finally the structure inducedon A by ( R , + , × ) is interdefinable with the structure induced on H ∩ [ , λ ) by ( R , + , × ) . So the structure induced on A by ( R , + , × ) is dp-minimal. (cid:3) The unique (up to sign) topological group isomorphism γ ∶ R / Z → G λ is γ ( t + Z ) = λ t − ⌊ t ⌋ . Given irrational α = s + Z ∈ R / Z we let χ α ∶ Z → G λ be χ α ( k ) ∶ = γ ( αk ) = λ sk − ⌊ sk ⌋ and let G α,λ be the structure induced on Z by ( R , + , × ) and χ α . Proposition 7.6.
Let α ∈ R / Z be irrational and λ > . Then G α,λ is a dp-minimalexpansion of ( Z , + , C α ) . Let S be the unit circle equipped with complex multiplication. The second familyof consists of S and other circle groups constructed from S in roughly the same waythat G λ is constructed from ( R > , × ) . We only discuss S . The unique (up to sign)topological group isomorphism γ ∶ R / Z → S is given by γ ( t + Z ) = e πit . Givenirrational α = s + Z ∈ R / Z we let χ α ∶ Z → S be χ α ( k ) ∶ = γ ( αk ) = e πisk . and let S α be the structure induced on Z by ( R , + , × ) and χ α . Proposition 7.7.
Let α ∈ R / Z be irrational. Then S α is a dp-minimal expansionof ( Z , + , C α ) . The third family comes from elliptic curves. Given an elliptic curve E defined over R we let E ( R ) be the real points of E . We consider E as a subvariety of P via theWeierstrass embedding. We let E ( R ) be the connected component of the identity of E ( R ) , so E ( R ) is a semialgebraic circle group. The fourth family of semialge-braic circle groups consists of such E ( R ) and circle groups constructed from E ( R ) in roughly the same way as G λ is constructed from ( R > , × ) . We only discuss E ( R ) .Fix λ > Z + iλ Z . Let E λ be the elliptic curve associ-ated to Λ, recall that E λ is defined over R and any elliptic curve defined over R isisomorphic to some E λ . Given η > E λ ( R ) → E η ( R ) if and only if λ / η ∈ Q , see [27].Let ℘ λ be the Weierstrass elliptic function associated to Λ and p λ ∶ R → E λ ( R ) begiven by p λ ( t ) = [ ℘ λ ( t ) ∶ ℘ ′ λ ( t ) ∶ ] . The unique (up to sign) topological groupisomorphism γ ∶ R / Z → E λ ( R ) is γ ( t + Z ) = p λ ( t ) . Fix irrational α = s + Z ∈ R / Z and let χ α ∶ Z → E λ ( R ) be the character χ α ( k ) ∶ = γ ( αk ) = p λ ( sk ) = [ ℘ λ ( sk ) ∶ ℘ ′ λ ( sk ) ∶ ] . Let E α,λ be the structure induced on Z by ( R , + , × ) and χ α . Proposition 7.8.
Let α ∈ R / Z be irrational and λ > . Then E α,λ is a dp-minimalexpansion of ( Z , + , C α ) . Another possible family of expansions.
We describe an approach to con-structing uncountably many dp-minimal expansions of each example describedabove. Let I be a closed bounded interval with interior. Let C ∞ ( I ) be the topo-logical vector space of smooth functions I → R where the topology is that inducedby the seminorms f ↦ max {∣ f ( n ) ( t )∣ ∶ t ∈ I } . So C ∞ ( I ) is a Polish space. LeGal has shown that the set of f ∈ C ∞ ( I ) such that ( R , + , × , f ) is o-minimal iscomeager [25]. Conjecture 2.
Let H be a semialgebraic Mordell-Lang circle group, γ be the unique(up to sign) topological group isomorphism R / Z → H , α ∈ R / Z be irrational, χ ∶ Z → H be given by χ ( k ) = γ ( αk ) , and A ∶ = χ ( Z ) . There is a comeager subset Λ of C ∞ ( I ) (possibly depending on α ) such that if f ∈ Λ then(1) ( R , + , × , f ) is o-minimal,(2) if f ≠ g are in Λ then ( R , + , × , f ) and ( R , + , × , g ) are not interdefinable.(3) Every ( R , + , × , f ) -definable group is definably isomorphic to a semialgebraicgroup and any ( R , + , × , f ) -definable homomorphism between semialgebraicgroups is semialgebraic.(4) ( R , + , × , A ) is NIP and Th ( R , + , × ) is an open core of Th ( R , + , × , A ) .(5) Every ( R , + , × , A ) -definable subset of A k is a finite union of sets of the form b ⊕ n ( X ∩ A k ) for semialgebraic X and b ∈ A k . So in particular the structureinduced on A by ( R , + , × ) is a dp-minimal expansion of ( Z , + , C α ) . Gorman, Hieronymi, and Kaplan generalized the Mordell-Lang property to an ab-stract model theoretic setting [14]. Item ( ) of Conjecture 2 should follow byverifying that the conditions in their paper are satisfied.Suppose Conjecture 2 holds. Let H α be the structure induced on Z by ( R , + , × ) and χ and for each f ∈ Λ let H α,f be the structure induced on Z by ( R , + , × , f ) and χ . So each H α,f is a dp-minimal expansion of H α . It is easy to see that our expansions of ( Z , + , C α ) define the same subsets of Z as ( Z , + , C α ) Sh , so is ( Z , + , C α ) Sh a reduct of these expansions? It is intuitivelyobvious that these expansions defines subsets of Z which are not definable in ( Z , + , C α ) Sh , but how do we show this? When are two of the expansions describedabove interdefinable? We now develop tools to answer these questions.8. The o-minimal completion
We associate an o-minimal expansion of ( R / Z , + , C ) to a strongly dependent ex-pansion of ( Z , + , C α ) . We will show that ( Z , + , C α ) Sh is interdefinable with thestructure induced on Z by ( R / Z , + , C ) and ψ α . It will follow that each of the dp-minimal expansions of ( Z , + , C α ) describe above in fact expands ( Z , + , C α ) Sh .We first recall the completion of an NIP expansion of a dense archimedean orderedabelian group defined in [49].8.1. The linearly ordered case.
Suppose that ( H, + , < ) is a dense subgroup of ( R , + , < ) , H is an expansion of ( H, + , < ) , and H ≺ N is highly saturated. Let Fin be the convex hull of H in N and Inf be the set of a ∈ N such that ∣ a ∣ < b forall positive b ∈ H . We identify Fin / Inf with R so the quotient map st ∶ Fin → R is the usual standard part map. Note that Fin and
Inf are both H Sh -definableso we regard R as an imaginary sort of N Sh . We let st ∶ Fin n → R n be given byst ( a , . . . , a n ) = ( st ( a ) , . . . , st ( a n )) . Fact 8.1 is [49, Theorem F].
Fact 8.1.
Suppose H is NIP . Then the following structures are interdefinable.(1) The structure H □ on R with an n -ary relation symbol defining the closurein R n of every subset of H n which is externally definable in H .(2) The structure on R with an n -ary relation symbol defining, for each N -definable subset X of N n , the image of Fin n ∩ X under the standard partmap Fin n → R n .(3) The open core of the structure induced on R by N Sh .Furthermore the structure induced on H by H □ is a reduct of H Sh . If H is stronglydependent then H □ is interdefinable with the structure induced on R by N Sh . The completion H □ should be “at least as tame” as H because H □ is interpretablein N Sh . In general H Sh is not interdefinable with the structure induced on H by H □ . Suppose H = R and H = ( R , + , < , Q ) , it follows from Theorem 9.2 andthe quantifier elimination for ( R , + , < , Q ) that ( R , + , < , Q ) □ is interdefinable with ( R , + , < ) . Recall that ( R , + , < , Q ) has dp-rank two [12]. We expect that if H is dp-minimal then H Sh is interdefinable with the structure induced on H by H □ . Notethat if H = R and H is dp-minimal then H is o-minimal by [40], so by the Marker-Steinhorn theorem H □ is the open core of H , so H □ and H are interdefinable asany o-minimal stucture is interdefinable with its open core.8.2. The cyclically ordered case.
We only work over ( Z , + , C α ) , but everythinggoes through for a cyclic order on an abelian group induced by an injective charac-ter to R / Z . Fix irrational α ∈ R / Z . Abusing notation we let ψ α ∶ Z n → ( R / Z ) n begiven by ψ α ( k , . . . , k n ) = ( αk , . . . , αk n ) . If β ∈ R / Z is irrational then C α = C β ifand only if α = β , so we can recover ψ α from ( Z , + , C α ) . Let Z ≺ N be highly saturated. We define a standard part map st ∶ N → R / Z bydeclaring st ( a ) to be the unique element of R / Z such that for all integers k, k ′ wehave C ( αk, st ( a ) , αk ′ ) if and only if C α ( k, a, k ′ ) . Note that st is a homomorphismand let Inf be the kernal of st. We identify N / Inf with R / Z and st with thequotient map. Note that Inf is convex, hence N Sh -definable. So we consider R / Z to be an imaginary sort of N Sh . Proposition 8.2.
Suppose Z is NIP . The following structures are interdefinable.(1) The structure Z □ on R / Z with an n -ary relation symbol defining the closurein ( R / Z ) n of ψ α ( X ) for every X ⊆ Z n which is externally definable in Z .(2) The structure on R / Z with an n -ary relation symbol defining the image ofeach N -definable X ⊆ N n under the standard part map N n → ( R / Z ) n .(3) The open core of the structure induced on R / Z by N Sh .Furthermore the structure induced on Z by Z □ and ψ α is a reduct of Z Sh . If Z isstrongly dependent then Z □ is interdefinable with the structure induced on R / Z by N Sh and Z □ is o-minimal. We expect that if Z is dp-minimal then the structure induced on Z by Z □ and ψ α isinterdefinable with Z Sh . All claims of Proposition 8.2 except o-minimality follow byslight modifications to the proof of Fact 8.1. The last claim also follows easily fromthe methods of [49], we provide details below. ( H need not be o-minimal when H is strongly dependent, for example ( Q , + , < , Z ) is strongly dependent by [12, 3.1]and ( Q , + , < , Z ) □ is interdefinable with ( R , + , < , Z ) .)We need three facts to prove the last claim. Fact 8.3 is left to the reader. Fact 8.3.
Suppose X is a subset of R / Z . Then X is a finite union of intervals andsingletons if and only if the boundary of X is finite. Fact 8.4 is essentially a theorem of Dolich and Goodrick [12]. They only treatlinearly ordered structures, but routine alternations to their proof yield Fact 8.4.
Fact 8.4.
Suppose ( G, + , S ) is a cyclically ordered abelian group, G is a stronglydependent expansion of ( G, + , S ) , and X is a G -definable subset of G . If X isnowhere dense then X has no accumulation points. Fact 8.5 follows from [49, Theorem B].
Fact 8.5.
Suppose Z is NIP and
X, Y are Z □ -definable subsets of ( R / Z ) n . Then X either has interior in Y or X is nowhere dense in Y . We now show that if Z is strongly dependent then Z □ is o-minimal. We let Bd ( X ) be the boundary of a subset X of R / Z . Proof.
Let Z be strongly dependent and X be an Z □ -definable subset of R / Z . ByFact 8.5 X is not dense and co-dense in any interval. So Bd ( X ) is nowhere dense.By Fact 8.4 Bd ( X ) has no accumulation points, so Bd ( X ) is finite by compactnessof R / Z . By Fact 8.3 X is a finite union of intervals and singletons. (cid:3) There is another way to show that Z □ is o-minimal when Z is dp-minimal. Suppose Z is dp-minimal. Then N Sh is dp-minimal, so Z □ is dp-minimal by Proposition 8.2.It follows from work of Simon [40] that an expansion of ( R / Z , + , C ) is o-minimal if and only if it is dp-minimal.In Section 8.3 we show ( Z , + , C α ) □ is interdefinable with ( R / Z , + , C ) and ( Z , + , C α ) Sh is interdefinable with the structure induced on Z by ( Z , + , C α ) □ and ψ α .8.3. The completion of ( Z , + , C α ) . Proposition 8.6 shows in particular that ( Q , + , < ) □ is the usual completion of ( Q , + , < ) . Proposition 8.6.
Suppose H is a dense subgroup of ( R , + ) . Then ( H, + , < ) □ isinterdefinable with ( R , + , < ) . Proposition 8.6 will require the quantifier elimination for archimedean orderedabelian groups. See Weispfennig [50] for a proof.
Fact 8.7.
Let ( H, + , < ) be an archimedean ordered abelian group. Then ( H, + , < ) admits quantifier elimination after adding a unary relation for every nH . We now prove Proposition 8.6. If T ∶ H n → H is a Z -linear function given by T ( a , . . . , a n ) = k a + . . . + k n a n for integers k , . . . , k n then we also let T denotethe function R n → R given by ( t , . . . , t n ) ↦ k t + . . . + k n t n . Proof.
Let ( H + < ) ≺ ( N, + , < ) be highly saturated and let Fin , st ∶ Fin n → R n be as above. As ( H, + , < ) is NIP, it suffices by Fact 8.1 to suppose that Y ⊆ N n is N -definable and show that st ( Y ∩ Fin n ) is ( R , + , < ) -definable. If Z is the closureof Y in N n then st ( Z ∩ Fin n ) = st ( Y ∩ Fin n ) . So we suppose that Y is closed.As Y is closed a straightforward application of Fact 8.7 shows that Y is a finiteunion of sets of the form { a ∈ N n ∶ T ( a ) ≤ s , . . . , T k ( a ) ≤ s k } for Z -linear T , . . . , T k ∶ N n → N and s , . . . , s k ∈ N . So we may suppose that Y is of this form. If s i > Fin then
Fin n is contained in { a ∈ N n ∶ T i ( a ) ≤ s i } and if s i < Fin then { a ∈ N n ∶ T i ( a ) ≤ s i } is disjoint from Fin n . So we suppose s , . . . , s k ∈ Fin . It is now easy to see thatst ( Y ∩ Fin n ) = { a ∈ R n ∶ T ( a ) ≤ st ( s ) , . . . , T k ( a ) ≤ st ( s k )} . So st ( Y ∩ Fin n ) is ( R , + , < ) -definable. (cid:3) Proposition 8.8 will be used to show that ( Z , + , C α ) Sh is interdefinable with thestructure induced on Z by ( R / Z , + , C ) and ψ α . Proposition 8.8.
Suppose H is a dense subgroup of ( R , + ) . Then ( H, + , < ) Sh isinterdefinable with the structure induced on H by ( R , + , < ) .Proof. As ( H, + , < ) is NIP Fact 8.1 shows that the structure induced on H by ( R , + , < ) is a reduct of ( H, + , < ) Sh . We show that ( H, + , < ) Sh is a reduct of thestructure induced on H by ( R , + , < ) . Suppose ( H, + , < ) ≺ ( N, + , < ) is highlysaturated and Y ⊆ N n is ( N, + , < ) -definable. Applying Fact 8.7 there is a family ( X ij ∶ ≤ i, j ≤ k ) of ( N, + , < ) -definable sets such that Y = ⋃ ki = ⋂ kj = X ij and each X ij is either ( N, + ) -definable or of the form { a ∈ N n ∶ T ( a ) ≤ s } for some Z -linear T ∶ N n → N and s ∈ N . As H n ∩ Y = ⋃ ki = ⋂ kj = ( H n ∩ X ij ) it is enough to showthat each H n ∩ X ij is definable in the structure induced on H by ( R , + , < ) . If X ij is ( N, + ) -definable then H n ∩ X ij is ( H, + ) -definable by stability of abelian groups, So suppose Y = { a ∈ N n ∶ T ( a ) ≤ s } for Z -linear T ∶ N n → N and s ∈ N . Let Fin and st be as above. If s > Fin then H n ⊆ Y and if s < Fin then H n is disjointfrom Y . Suppose s ∈ Fin . If s ≥ st ( s ) then H n ∩ Y = { a ∈ H n ∶ T ( a ) ≤ st ( s )} and if s < st ( s ) then H n ∩ Y = { a ∈ H n ∶ T ( a ) < st ( x )} . So in each case H n ∩ Y is definable in the structure induced on H by ( R , + , < ) . (cid:3) We can now compute ( Z , + , C α ) □ . Proposition 8.9.
Fix irrational α ∈ R / Z . Then ( Z , + , C α ) □ is interdefinable with ( R / Z , + , C ) and ( Z , + , C α ) Sh is interdefinable with the structure induced on Z by ( R / Z , + , C ) and ψ α .Proof. Let π be the quotient map R → R / Z so ( R , + , < , , π ) is a universal coverof ( R / Z , + , C ) . Fix λ ∈ R such that α = λ + Z , let H ∶ = Z + λ Z , and let ρ ∶ H → Z be ρ ∶ = ψ − α ◦ π , so that ( H, + , < , , ρ ) is a universal cover of ( Z , + , C α ) . Let ρ ∶ H n → Z n be given by ρ ( t , . . . , t n ) = ( ρ ( t ) , . . . , ρ ( t n )) . Suppose X ⊆ Z n is ( Z , + , C α ) Sh -definable. Then Y ∶ = ρ − ( X ) ∩ [ , ) n is easily seen to be externallydefinable in ( H, + , < ) . Proposition 8.6 shows that Cl ( Y ) is ( R , + , < ) -definable.Observe that π ( Cl ( Y )) is the closure of ψ α ( X ) in ( R / Z ) n . So the closure of ψ α ( X ) in ( R / Z ) n is definable in ( R / Z , + , C ) . So ( Z , + , C α ) □ is interdefinable with ( R / Z , + , C ) .We now show that ( Z , + , C α ) Sh is interdefinable with the structure induced on Z by ( R / Z , + , C ) and ψ α . By Proposition 8.2 and preceding paragraph it suffices toshow that ( Z , + , C α ) Sh is a reduct of the induced structure. Again suppose that X ⊆ Z n is an ( Z , + , C α ) Sh -definable subset of Z n and Y ∶ = ρ − ( X ) ∩ [ , ) n . ByProposition 8.8 Y is definable in the structure induced on H ∩ [ , ) by ( R , + , < ) .So ρ ( Y ) = X is definable in the structure induced on ψ α ( Z ) by ( R / Z , + , C ) . Hence Y is definable in the structure induced on Z by ( R / Z , + , C ) and ψ α . (cid:3) Corollary 8.10 now follows immediately, we leave the details to the reader.
Corollary 8.10.
Suppose H is a semialgebraic Mordell-Lang circle group, γ is theunique (up to sign) topological group isomorphism R / Z → H , α ∈ R / Z is irrational, χ ∶ Z → H is the character χ ( k ) ∶ = γ ( αk ) , and H α is the structure induced on Z by ( R , + , × ) and χ . Then H α expands ( Z , + , C α ) Sh . So in particular G α,λ , S α , and E α,η all expand ( Z , + , C α ) Sh for any λ, η > . By Fact 6.6 a dp-minimal expansion of ( Z , + , C α ) Sh cannot add new unary sets.We suspect that any dp-minimal expansion of ( Z , + , C α ) Sh adds new binary sets. Proposition 8.11.
Fix irrational α ∈ R / Z . Then G α,λ , S α , and E α,η all define asubset of Z which is not ( Z , + , C α ) Sh -definable for any λ, η > . An open subset of a topological space is regular if it is the interior of its closure.
Proof.
We treat G α,λ , the other cases follow in the same way. Let S the cyclic orderon [ , λ ) where S ( t, t ′ , t ′′ ) if and only if either t < t ′ < t ′′ , t ′ < t ′′ < t or t ′′ < t < t ′ . So S is the unique (up to opposite) semialgebraic cyclic group order on ([ , λ ) , ⊗ λ ) .Let U be a regular open semialgebraic subset of [ , λ ) which is not definable in ([ , λ ) , ⊗ λ , S ) , e.g. an open disc contained in [ , λ ) . Let V ∶ = χ − α ( U ) , so V is G α,λ -definable. Suppose that V is ( Z , + , C α ) Sh -definable. By Proposition 8.9 the closure of χ α ( V ) is definable in ([ , λ ) , ⊗ λ , S ) . As χ α ( Z ) is dense in [ , λ ) , theclosure of χ α ( V ) agrees with the closure of U . As U is regular U is the interior ofthe closure of U . So U is definable in ([ , λ ) , ⊗ λ , S ) , contradiction. (cid:3) When the examples are interdefinable
In this section we describe the completions of the dp-minimal expansions of ( Z , + , C α ) constructed in Section 7 and show that if two of these expansions are interdefinablethen the associated semialgebraic circle groups are semialgebracially isomorphic.Suppose R is an o-minimal expansion of ( R , + , × ) , H is an R -definable circle group.We say that a subgroup A of H is a GH-subgroup if ( R , A ) is NIP, Th ( R ) is anopen core of Th ( R , A ) , and the structure induced on A by R is dp-minimal. Proposition 9.1.
Suppose R is an o-minimal expansion of ( R , + , × ) , H is an R -definable circle group, γ is the unique (up to sign) topological group isomorphism R / Z → H , χ is an injective character Z → H , and Z is the structure induced on Z by R and χ . If χ ( Z ) is a GH -subgroup then Z □ is interdefinable with the structureinduced on R / Z by R and γ . So for any irrational α ∈ R / Z and λ > :(1) A subset of ( R / Z ) n is G □ α,λ -definable if and only if it is the image underthe quotient map R n → ( R / Z ) n of a set of the form {( t , . . . , t n ) ∶ ( λ t , . . . , λ t n ) ∈ X } for a semialgebraic subset X of [ , λ ) n .(2) A subset of ( R / Z ) n is S □ α -definable if and only if it is the image under thequotient map R n → ( R / Z ) n of a set of the form {( t , . . . , t n ) ∈ [ , ) n ∶ ( e πit , . . . , e πit n ) ∈ X } for a semialgebraic subset X of S n .(3) A subset of ( R / Z ) n is E □ α,λ -definable if and only if it is an image under thequotient map R n → ( R / Z ) n of a set of the form {( t , . . . , t n ) ∈ [ , ) n ∶ ( p λ ( t ) , . . . , p λ ( t n )) ∈ X } for a semialgebraic subset X of E λ ( R ) n . It follows from Proposition 9.1 that if Z is one of the expansions of ( Z , + , C α ) de-scribed above then Z □ defines an isomorphic copy of ( R , + , × ) , so if Z ≺ N is highlysaturated then N Sh interprets ( R , + , × ) . So Z is non-modular. An adaptation of[49, Proposition 15.2] shows that N cannot interpret an infinite field.We prove Theorem 9.2, a more general result on completions which covers almostall “dense pairs”. It is easy to see that Proposition 9.1 follows from Theorem 9.2,we leave the details of this to the reader. Theorem 9.2.
Let S be an o-minimal expansion of ( R , + , < ) . Suppose A is asubset of R m such that ( S , A ) is NIP and Th ( S ) is an open core of Th ( S , A ) . Let A be the structure induced on A by S and X be the closure of A in R m . Then(1) the structure A □ with domain X and an n -ary relation symbol defining Cl ( Y ) for each A Sh -definable Y ⊆ A n .(2) and the structure X induced on X by S ,are interdefinable. (Note that X is S -definable.) We let ∥ a ∥ ∶ = max {∣ a ∣ , . . . , ∣ a n ∣} for all a = ( a , . . . , a n ) ∈ R n . We will need a metric argument from [49] to show that X is a reduct of A □ . If X = R m then one can can give a topological proof following [48, Proposition 3.4]. Proof.
We first show that X is a reduct of A □ . Suppose Y is a nonempty S -definablesubset of X n . By o-minimal cell decomposition there are definable closed subsets E , F . . . , E k , F k of R nm such that Y = ⋃ ki = ( E i \ F i ) . We have Y = k ⋃ i = (( X n ∩ E i ) \ ( X n ∩ F i )) so we may suppose that Y is a nonempty closed S -definable subset of X n . Let W be the set of ( a, a ′ , c ) ∈ X × X × X n for which there is c ′ ∈ Y satisfying ∥ c − c ′ ∥ < ∥ a − a ′ ∥ . So W ∩ ( A × A × A n ) is A -definable and Z ∶ = Cl ( W ∩ ( A × A × A n )) is A □ -definable. The metric argument in the proof of [49, Lemma 13.5] shows that Y = ⋂ a,a ′ ∈ X,a ≠ a ′ { c ∈ X ∶ ( a, a ′ , c ) ∈ Z } . (This metric argument requires Y to be closed.) So Y is A □ -definable.We now show that X is a reduct of A □ . Suppose Y is an A Sh -definable subset of A n . We show that Cl ( Y ) ⊆ X n is S -definable. As ( S , A ) is NIP, A is NIP, so anapplication of Fact 3.5 yields an A -definable family ( Y a ∶ a ∈ A k ) of subsets of A n such that for every finite B ⊆ Y we have B ⊆ Y a ⊆ Y for some a ∈ A k . As Th ( S ) isan open core of Th ( S , A ) there is an S -definable family ( Z b ∶ b ∈ R l ) of subsets of R nm such that for every a ∈ A k we have Cl ( Y a ) = Z b for some b ∈ R l . So for everyfinite F ⊆ X there is b ∈ R l such that F ⊆ Z b ⊆ Cl ( Y ) . A saturation argumentyields an R Sh -definable subset Z of X n such that Y ⊆ Z ⊆ Cl ( Y ) . An applicationof Fact 3.6 shows that Z is S -definable, so Cl ( Z ) = Cl ( Y ) is S -definable. (cid:3) The proof of Theorem 9.2 goes through for any expansion S of ( R , + , < ) suchthat S is NIP, every S -definable set is a boolean combination of definable closedsets, and S Sh is interdefinable with S . So for example Theorem 9.2 holds when S = ( R , + , < , Z ) .Our next goal is to show that if H and Z are as in Proposition 9.1 then we canrecover R and H from Z . We show that we can recover R and H from Z □ . Thisfollows from a general correspondence between(1) non-modular o-minimal expansions C of ( R / Z , + , C ) , and(2) pairs of the form ⟨ R , H ⟩ , for an o-minimal expansion R of ( R , + , × ) and an R -definable circle group H .In this correspondence C is unique up to interdefinibility, R is unique up to inter-definibility, and H is unique up to R -definable isomorphism.Suppose that R is an o-minimal expansion of ( R , + , × ) and H is an R -definable cir-cle group. We consider H as a topological group with T H . Let γ be the unique (upto sign) topological group isomorphism R / Z → H . Let C be the structure inducedon R / Z by R and γ . Note C is unique up to interdefinibility. It is easy to see that C defines an isomorphic copy of ( R , + , × ) .Now suppose C is a non-modular o-minimal expansion of ( R / Z , + , C ) . Suppose I is a non-empty open interval and ⊕ , ⊗ ∶ I → I are C -definable such that ( I, ⊕ , ⊗ ) is isomorphic to ( R , + , × ) . Let ι be the unique isomorphism ( R , + , × ) → ( I, ⊕ , ⊗ ) .Let R be the structure induced on R by C and ι . By compactness of R / Z there is afinite A ⊆ R / Z such that ( a + I ∶ a ∈ A ) covers R / Z . Fix a bijection f ∶ B → A forsome B ⊆ R . Let τ ∶ B × R → R / Z be the surjection given by τ ( b, t ) = f ( b ) + ι ( t ) .Observe that equality modulo τ is an R -definable equivalence relation and, applyingdefinable choice, let H be an R -definable subset of B × R which contains one elementfrom each fiber of τ . Let τ ′ ∶ H → R / Z be the induced bijection and ⊞ be thepullback of + by τ ′ . Then H ∶ = ( H, ⊞ ) is an R -definable circle group. Note thatthe expansion of ( R / Z , + , C ) associated to ⟨ R , H ⟩ is interdefinable with C . Proposition 9.3.
For i ∈ { , } suppose that R i is an o-minimal expansion of ( R , + , × ) , H i is an R i -definable circle group, and C i is the expansion of ( R / Z , + , C ) associated to ⟨ R i , H i ⟩ . If C and C are interdefinable then R and R are interde-finable and there is an R -definable group isomorphism H → H .Proof. It is easy to see that C is bi-interpretable with R and C is bi-interpretablewith R . So if C and C are interdefinable then R and R are bi-interpretable,hence interdefinable by Fact 5.3. So we suppose R = R and denote R by R .For each i ∈ { , } let I i be an interval in R / Z and ι i be a bijection R → I i suchthat R is interdefinable with the structure induced on R by C i and ι i . Let F i bethe pushforward of R by ι i for i ∈ { , } . So F is a C -definable copy of R and F is a C -definable copy of R . Let H , H be the pushforward of H , H by ι , re-spectively. Likewise, let H , H be the pushforward of H , H by ι , respectively.So H , H are F -definable copies of H , H , respectively, and H , H are F -definable copies of H , H , respectively. Given i ∈ { , } let γ i be a C i -definablegroup isomorphism H i → R / Z . (Note that C a priori does not define a groupisomorphism from H to R / Z , likewise for C and H .)Now suppose that C and C are interdefinable. We show that H and H are R -definably isomorphic. It suffices to show that H and H are F -definablyisomorphic. As F and C are bi-interpretable it is enough to produce a C -definablegroup isomorphism H → H . As C and C are interdefinable γ − ◦ γ is a C -definable group isomorphism H → H . By Fact 5.3 there is a C -definablebijection ξ ∶ I → I which induces an isomorphism (up to interdefinibility) from F to F . Let ζ be the C -definable group isomorphism H → H induced by ξ .Then ζ ◦ γ ◦ γ − is a C -definable group isomorphism H → H . (cid:3) Theorem 9.4 classifies our examples up to interdefinibility.
Theorem 9.4.
Let R , R , H , H be as in Proposition 9.3. Fix irrational α ∈ R / Z .For each i ∈ { , } let γ i ∶ R / Z → H i be the unique (up to sign) topological groupisomorphism, χ i ∶ Z → H i be given by χ i ( k ) ∶ = γ i ( αk ) , and Z i be the structureinduced on Z by R i and χ i . Suppose χ i ( Z ) is a GH -subgroup for i ∈ { , } . Then Z and Z are interdefinable if and only if R and R are interdefinable and thereis an R -definable group isomorphism H → H . Proof.
Suppose that R and R are interdefnable and ξ ∶ H → H is an R -definable group isomorphism. Then ξ ◦ γ is the unique (up to sign) topologicalgroup isomorphism R / Z → H . So after possibly replacing ξ with − ξ we have γ = ξ ◦ γ , hence χ = ξ ◦ χ . It easily follows that Z and Z are interdefinable.Suppose Z and Z are interdefinable. Then Z □ and Z □ are interdefinable. ByProposition 9.1 the expansions of ( R / Z , + , C ) associated to ⟨ R , H ⟩ and ⟨ R , H ⟩ are interdefinable. Applying Proposition 9.3 see that R and R are interdefinableand there is an R -definable group isomorphism H → H . (cid:3) We now see that we have constructed uncountably many dp-minimal expansions ofeach ( Z , + , C α ) Sh . Corollary 9.5 follows from Theorem 9.4 and the classification ofone-dimensional semialgebraic groups described above. Corollary 9.5.
Fix irrational α ∈ R / Z and let λ, η > . Then(1) no two of G α,λ , S α , E α,η are interdefinable,(2) G α,λ and G α,η are interdefinable if and only if λ / η ∈ Q ,(3) E α,λ and E α,η are interdefinable if and only if λ / η ∈ Q . Suppose for the rest of this section that Conjecture 2 holds. Fix irrational α ∈ R / Z .Suppose that H is a semialgebraic Mordell-Lang circle group, γ ∶ R / Z → H is theunique (up to sign) topological group isomorphism, and χ ∶ Z → H is the character χ ( k ) ∶ = γ ( αk ) . Let H α be the structure induced on Z by ( R , + , × ) and χ . Forany f ∈ Λ let H α,f be the structure induced on Z by ( R , + , × , f ) and χ . Then H α,f is dp-minimal and H □ α,f is interdefinable with the structure induced on R / Z by ( R , + , × , f ) and γ . It follows that by Proposition 9.1 that H α,f is a properexpansion of H α and if f, g are distinct elements of Λ then H α,f and H α,g are notinterdefinable.In this way, still assuming Conjecture 2, we can produce produce two dp-minimalexpansions of H α which do not have a common NIP expansion. Let h ∈ C ∞ ( I ) be such that ( R , + , × , h ) is not NIP. (For example one can arrange that I = [ , ] and { t ∈ I ∶ f ( t ) = } is 0 ∪ { / n ∶ n ≥ } .) As Λ is comeager an application of thePettis lemma [22, Theorem 9.9] implies that there are f, g ∈ Λ and t > f − g = th . So after rescaling h we suppose f − g = h . Suppose that Z is an NIPexpansion of both H α,f and H α,g . Then Z □ is NIP. An easy argument using thefirst part of the proof of Theorem 9.2 shows that ( R , + , × , f, g ) is interpretable in Z □ , contradiction. (This kind of argument was previously used by Le Gal [25] toshow that there are two o-minimal expansions of ( R , + , × ) which are not reductsof a common o-minimal structure.)10. Dp-minimal expansions of ( Z , + , Val p ) Throughout p is a fixed prime. To avoid mild technical issues we assume p ≠ Z p .)10.1. A proper dp-minimal expansion of ( Z , + , Val p ) . We apply work of Mari-aule. The first and third claims of Fact 10.1 are special cases of the results of [29].The second claim follows from Mariaule’s results and a general theorem of Boxalland Hieronymi on open cores [6]. Recall that 1 + p Z p is a subgroup of Z × p . Fact 10.1.
Suppose that A is a finitely generated dense subgroup of ( + p Z p , × ) .Then ( Z p , + , × , A ) is NIP , Th ( Z p , + , × ) is an open core of Th ( Z p , + , × , A ) , andevery ( Z p , + , × , A ) -definable subset of A k is of the form X ∩ Y where X is an ( A, × ) -definable subset of A k and Y is a semialgebraic subset of Z kp . We let Exp be the p -adic exponential, i.e.Exp ( a ) ∶ = ∞ ∑ n = a n n ! for all a ∈ p Z p . (The sum does not converge off p Z p .) Exp is a topological group isomorphism ( p Z p , + ) → ( + p Z p , × ) . So a ↦ Exp ( pa ) is a topological group isomorphism ( Z p , + ) → ( + p Z p , × ) . It is easy to see thatVal p ( Exp ( a ) − ) = Val p ( a ) for all a ∈ p Z p . So for all a ∈ Z p we haveVal p ( Exp ( pa ) − ) = Val p ( pa ) = Val p ( a ) + . Define v ( b ) = Val p ( b − ) − b ∈ + p Z p , so a ↦ Exp ( pa ) is an isomorphism ( Z p , + , Val p ) → ( + p Z p , × , v ) .We let χ ∶ Z → Z × p be the character χ ( k ) ∶ = Exp ( pk ) and let P be the structureinduced on Z by ( Z p , + , × ) and χ . Note that P expands ( Z , + , Val p ) because χ isan isomorphism ( Z , + , Val p ) → ( χ ( Z ) , × , v ) . Let χ ( k , . . . , k n ) = ( χ ( k ) , . . . , χ ( k n )) for all ( k , . . . , k n ) ∈ Z n . There are P -definable subsets of Z which are not ( Z , + , Val p ) -definable. Consider N ∪ { ∞ } as the value set of ( Z , + , Val p ) . It follows from the quantifier elimination for ( Z , + , Val p ) that the structure induced on N ∪ { ∞ } by ( Z , + , Val p ) is interdefinablewith ( N ∪ { ∞ } , < ) . So Val p − ( N ) is P -definable and not ( Z , + , Val p ) -definable.Let E be the set of a ∈ + p Z p such that v ( a ) ∈ N . Note that if b, c ∈ + p Z p then χ − ( bE ) = χ − ( cE ) if and only if b = c . So P defines uncountably many subsets of Z , in constrast ( Z , + , Val p ) defines only countably many subsets of Z . Proposition 10.2. P is dp-minimal. Proposition 10.2 requires some preliminaries. A formula ϑ ( x ; y ) is bounded if P ⊧ ∀ y ∃ ≤ n xϑ ( x ; y ) for some n . Let L Ab be the language of abelian groups togetherwith unary relations ( D n ) n ≥ and L Val be the expansion of L Ab by a binary relation ≼ p We let D n define n Z and declare k ≼ p k ′ if and only if Val p ( k ) ≤ Val p ( k ′ ) .Fact 10.3 was independently proven by Alouf and d’Elb´ee [2], Mariaule [28], andGuignot [15]. Fact 10.3. ( Z , + , Val p ) has quantifier elimination in L Val . We let L In be the language with an n -ary relation symbol defining χ − ( X ) for eachsemialgebraic X ⊆ Z np . So Y ⊆ Z n is quantifier free L In -definable if and only if Y = χ − ( X ) for semialgebraic X ⊆ Z np . Take P to be an ( L Val ∪ L In ) -structure.We first give a description of unary P -definable sets. Lemma 10.4.
Suppose ϕ ( x ; y ) , ∣ x ∣ = is an ( L Val ∪ L In ) -formula. Then ϕ ( x ; y ) isequivalent to a finite disjunction of formulas of the form ϕ ( x ; y ) ∧ ϕ ( x ; y ) where ϕ ( x ; y ) is a quantifier free L In -formula and ϕ ( x ; y ) is an L Ab -formula such thateither ϕ ( x ; y ) is bounded or there are integers k, l such that Val p ( k ) = and forevery b ∈ Z ∣ y ∣ , ϕ ( Z ; b ) = ( k Z + l ) \ A for finite A . The condition Val p ( k ) = ϕ ( Z ; b ) is dense in the p -adic topology. Proof.
By Fact 10.1 we may suppose ϕ ( x ; y ) = ϕ ( x ; y ) ∧ ϕ ( x ; y ) where ϕ is an L Val -formula and ϕ is a quantifier free L In -formula. By Fact 10.3 we may suppose ϕ ( x ; y ) = m ⋁ i = m ⋀ j = ϑ ij ( x ; y ) where each ϑ ij ( x ; y ) is an atomic L Val -formula. So we may suppose ϕ ( x ; y ) = m ⋁ i = ( ϕ ( x ; y ) ∧ m ⋀ j = ϑ ij ( x ; y )) . So we suppose ϕ ( x ; y ) is of the form ϕ ( x ; y ) ∧ ⋀ mj = ϑ j ( x ; y ) where each ϑ j ( x ; y ) isan atomic L Val -formula. After possibly rearranging there is 0 ≤ m ′ ≤ m such that(1) if 1 ≤ j ≤ m ′ then ϑ j ( x ; y ) is of the form g ≼ p h or g ≺ p h where g, h are L Ab -terms in the variables x, y , and(2) if m ′ < j ≤ m then ϑ j ( x ; y ) is an atomic L Ab -formula.Note that any formula of type ( ) is equivalent to a quantifier free L In -formula.Now ⎛⎜⎝ ϕ ( x ; y ) ∧ m ′ ⋀ j = ϑ j ( x ; y )⎞⎟⎠ ∧ m ⋀ j = m ′ + ϑ j ( x ; y ) . The formula inside the parentheses is equivalent to a quantifier free L In -formula.So we suppose that ϕ ( x ; y ) is for the form ϕ ( x ; y ) ∧ ϕ ( x ; y ) where ϕ ( x ; y ) is an L Ab -formula and ϕ ( x ; y ) is an quantifier free L In -formula. An easy application ofquantifier elimination shows that ϕ ( x ; y ) is equivalent to a formula of the form ⋁ mi = θ i ( x ; y ) where for each i either:(1) θ i ( x ; y ) is bounded, or(2) there are integers k ≠ , l and an L Ab -formula θ ′ i ( x ; y ) such that θ i ( x ; y ) isequivalent to ( x ∈ k ′ Z + l ) ∧ ¬ θ ′ i ( x ; y ) and θ ′ i ( x ; y ) is bounded.Applying the same reasoning as above we may suppose that ϕ ( x ; y ) satisfies ( ) or ( ) above. If ϕ ( x ; y ) is bounded then we are done. So fix integers k ′ , l andbounded ϕ ′ ( x ; y ) such that ϕ ( x ; y ) is equivalent to ( x ∈ k ′ Z + l ) ∧ ¬ ϕ ′ ( x ; y ) . Let v ∶ = Val p ( k ′ ) and k ∶ = k ′ / p v . So k ′ Z + l = ( p v Z + l ) ∩ ( k Z + l ) and ϕ ( x ; y ) islogically equivalent to [ Val p ( x − l ) ≥ v ] ∧ [ x ∈ k Z + l ] ∧ ¬ ϕ ′ ( x ; y ) . After replacing ϕ ( x ; y ) with [ x ∈ k Z + l ] ∧ ¬ ϕ ′ ( x ; y ) and replacing ϕ ( x ; y ) with [ Val p ( x − l ) ≥ v ] ∧ ϕ ( x ; y ) we may suppose that for every b ∈ Z ∣ y ∣ , ϕ ( Z ; b ) agreeswith ( k Z + l ) \ A for finite A . (cid:3) We also need Fact 10.5, a consequence of the quantifier elimination for ( Z p , + , × ) . Fact 10.5.
Suppose that φ ( x ; y ) , ∣ x ∣ = is a formula in the language of rings.Then there are formulas φ ( x ; y ) , φ ( x ; y ) such that(1) φ ( x ; y ) and φ ( x ; y ) ∨ φ ( x ; y ) are equivalent in ( Z p , + , × ) ,(2) φ ( Z p ; b ) is finite and φ ( Z p ; b ) is open for every b ∈ Z ∣ y ∣ p . Lemma 10.6 follows from inp-minimality of ( Z p , + , × ) and the fact that χ ( X ) isdense in 1 + p Z p . We leave the verification to the reader. Lemma 10.6.
Suppose that ϕ ( x ; y ) , φ ( x ; y ) , ∣ x ∣ = are quantifier free L In -formulassuch that ϕ ( Z ; b ) and φ ( Z ; b ) are both open in the p -adic topology for every b ∈ Z ∣ y ∣ .Then ϕ ( x ; y ) and φ ( x ; y ) cannot violate inp-minimality. We now prove Proposition 10.2.
Proof.
We equip Z with the p -adic topology. Fact 10.1 shows that P is NIP soit is enough to show that P is inp-minimal. Suppose towards a contradiction that ϕ ( x ; y ) , φ ( x ; z ) , and n violate inp-minimality. Applying Lemma 10.4, Fact 3.1, andFact 3.2 we may suppose there are ϕ ( x ; y ) , ϕ ( x ; y ) and k, l such that(1) ϕ ( x ; y ) = ϕ ( x ; y ) ∧ ϕ ( x ; y ) ,(2) ϕ ( x ; y ) is a quantifier free L In -formula, and(3) Val p ( k ) = b ∈ Z ∣ y ∣ , ϕ ( Z ; b ) = ( k Z + l ) \ A for finite A .Applying Fact 10.5 we get L In -formulas ϕ ′ ( x ; y ) and ϕ ′′ ( x ; y ) such that ϕ ′ ( x ; y ) is bounded, ϕ ′′ ( Z ; b ) is open for all b ∈ Z ∣ y ∣ , and ϕ ( x ; y ) = ϕ ′ ( x ; y ) ∨ ϕ ′′ ( x ; y ) .Applying Facts 3.1 and 3.2 we may suppose that ϕ ( Z ; b ) is open for all b ∈ Z ∣ y ∣ .We have reduced to the case when ϕ ( x ; y ) = ϕ ( x ; y ) ∧ ϕ ( x ; y ) where ϕ ( x ; y ) isa quantifier free L In -formula such that each ϕ ( Z ; b ) is open and there are k, l suchthat Val p ( k ) = b ∈ Z ∣ y ∣ we have ϕ ( Z ; b ) = ( k Z + l ) \ A for finite A .By the same reasoning we may suppose that there are φ ( x ; z ) , φ ( x ; z ) and k ′ , l ′ which satisfy the same conditions with respect to φ ( x ; z ) .We show that ϕ ( x ; y ) , φ ( x ; z ) and n violate inp-minimality and thereby obtain acontradiction with Lemma 10.6. Fix a , . . . , a m ∈ Z ∣ y ∣ and b , . . . , b m ∈ Z ∣ z ∣ suchthat ϕ ( x ; a ) , . . . , ϕ ( x ; a m ) and φ ( x ; b ) , . . . , φ ( x ; b m ) are both n -inconsistent and P ⊧ ∃ xϕ ( x ; a i ) ∧ φ ( x ; a j ) for all i, j . So P ⊧ ∃ xϕ ( x ; a i ) ∧ φ ( x ; b j ) for all i, j . Itsuffices to show that ϕ ( x ; a ) , . . . , ϕ ( x ; a m ) and φ ( x ; b ) , . . . , φ ( x ; b m ) are both n -inconsistent. We prove this for ϕ , the same argument works for φ . Fix a subset I of { , . . . , m } such that ∣ I ∣ = n . Let U ∶ = ⋂ i ∈ I ϕ ( Z ; a i ) and F ∶ = ⋂ i ∈ I ϕ ( Z ; a i ) .So F ∩ U is empty as ϕ ( x ; a ) , . . . , ϕ ( x ; a m ) is n -inconsistent. Observe that U isopen and F = ( k Z + l ) \ A for finite A . So F is dense in Z as Val p ( k ) =
0. So F ∩ U is the intersection of a dense set and an open set, so U is empty. Thus ϕ ( x ; a ) , . . . ϕ ( x ; a m ) is n -inconsistent. (cid:3) The p -adic completion Among other things we show that ( Q p , + , × ) is interpretable in the Shelah expan-sion of a highly saturated elementary extension of P , so P is non-modular.We construct a p -adic completion Z □ of a dp-minimal expansion Z of ( Z , + , Val p ) .We show that Z □ is dp-minimal, but in contrast with the situation over ( Z , + , C α ) we do not obtain an explicit description of unary definable sets. So we first showthat definable sets and functions in dp-minimal expansions of ( Z p , + , Val p ) behavesimilarly to definable sets and functions in o-minimal structures.11.1. Dp-minimal expansions of ( Z p , + , Val p ) . Let Y expand ( Z p , + , Val p ) . Fact 11.1.
Suppose Y is dp-minimal. Then the following are satisfied for any Y -definable subset X of Z np and Y -definable function f ∶ X → Z mp .(1) X is a boolean combination of Y -definable closed subsets of Z np .(2) If n = then X is the union of a definable open set and a finite set.(3) The dp-rank of X , the acl -dimension of X , and the maximal ≤ d ≤ n for which there is a coordinate projection π ∶ Z mp → Z dp such that π ( X ) hasinterior are all equal. (We denote the resulting dimension by dim X .)(4) There is a Y -definable Y ⊆ X such that dim X \ Y < dim X and f iscontinuous on Y .(5) The frontier inequality holds, i.e. dim Cl ( X ) \ X < dim X .Furthermore the same properties hold in any elementary extension of Y . Fact 11.1 is a special case of the results of [43]. Every single item of Fact 11.1 failsin ( Z , + , Val p ) because of the presence of dense and co-dense definable sets.There are dp-minimal expansions of valued groups in which algebraic closure doesnot satisfy the exchange property [5, 23], but this cannot happen over ( Z p , + , Val p ) Proposition 11.2.
Suppose Y is dp-minimal. Then Y is a geometric structure, i.e. Y eliminates ∃ ∞ and algebraic closure satisfies the exchange property.Proof. Elimination of ∃ ∞ follows from Fact 6.8. We show that algebraic closuresatisfies exchange. By [43, Proposition 5.2] exactly one of the following is satisfied,(1) algebraic closure satisfies exchange, or(2) there is definable open U ⊆ Z p , definable F ⊆ U × Z p such that each F a isfinite, for every a ∈ U there is an open a ∈ V ⊆ U such that F b = F a for all b ∈ V , and the family ( F a ∶ a ∈ U ) contains infinitely many distinct sets.Suppose ( ) holds. Let E be the set of ( a, b ) ∈ U such that F a = F b . Then E is adefinable equivalence relation, every E -class is open, and there are infinitely many E -classes. Suppose A ⊆ U contains exactly one element from each E -class. As Z p is separable ∣ A ∣ = ℵ . Let D ∶ = ⋃ a ∈ U F a = ⋃ a ∈ A F a . So D ⊆ Z p is definable and ∣ D ∣ = ℵ . This contradicts Fact 11.1 ( ) . (cid:3) Finally, Fact 11.3 is proven in [42].
Fact 11.3.
A dp-minimal expansion of ( Z p , + , × ) is ( Z p , + , × ) -minimal. It is an open question whether the theory of a dp-minimal expansion of ( Z p , + , × ) is Th ( Z p , + , × ) -minimal (equivalently: P -minimal).11.2. The p -adic completion. Suppose Z is an expansion of ( Z , + , Val p ) . Let S ≺ N be highly saturated. We define a standard part map st ∶ N → Z p bydeclaring st ( a ) to be the unique element of Z p such that for all non-zero integers k, k ′ we have Val p ( a − k ) ≥ k ′ if and only if Val p ( st ( a ) − k ) ≥ k ′ . Note that st isa homomorphism and let Inf be the kernal of st. We identify N / Inf with Z p andidentify st with the quotient map. Note that Inf is the set of a ∈ N such that st ( a ) ≥ k for all integers k , so Inf is externally definable and we consider Z p as animaginary sort of N Sh . Proposition 11.4.
Suppose Z is NIP . Then the following are interdefinable.(1) The structure Z □ on Z p with an n -ary relation symbol defining the closurein Z np of every Z Sh -definable subset of Z n .(2) The structure on Z p with an n -ary relation symbol defining the image ofeach N -definable subset of N n under the standard part map N n → Z np .(3) The open core of the structure induced on Z p by N Sh .The structure induced on Z by Z □ is a reduct of Z Sh . If Z is dp-minimal then Z □ is interdefinable with the structure induced on Z p by N Sh . So in particular Z □ is dp-minimal when Z is dp-minimal. All claims of Proposi-tion 11.4 except the last follow by easy alternations to the proof of Fact 8.1.We prove the last claim of Proposition 11.4. Proof.
Suppose Z is dp-minimal. We show that Z □ is interdefinable with the struc-ture induced on Z p by N Sh . It suffices to show that the induced structure on Z p is interdefinable with its open core. The structure induced on Z p by N Sh is dp-minimal as N Sh is dp-minimal. So by Fact 11.1 any N Sh -definable set is a booleancombination of closed N Sh -definable sets. (cid:3) One can show that ( Z , + , Val p ) □ is interdefinable with ( Z p , + , Val p ) . We omit thisfor the sake of brevity.We now give the p -adic analogue of Theorem 9.2. The proof is essentially the sameas that of Theorem 9.2 so we leave the details to the reader. (One applies Fact 3.7at the same point that Fact 3.6 is applied in the proof of Theorem 9.2.) Proposition 11.5.
Suppose that A is a subset of Z np , ( Z p , + , × , A ) is NIP , and Th ( Z p , + , × ) is an open core of Th ( Z p , + , × , A ) . Let A be the structure induced on A by ( Z p , + , × ) and X be the closure of A in Z np . Then(1) The structure A □ with domain X and an n -ary relation for the closure in X n of each A Sh -definable subset of A n ,(2) and the structure X induced on X by ( Z p , + , × ) ,are interdefinable. (Note that X is semialgebraic.) Fact 10.1 and Proposition 11.5 together easily yield Proposition 11.6.
Proposition 11.6.
The completion P □ of P is interdefinable with the structureinduced on Z p by ( Z p , + , × ) and a ↦ Exp ( pa ) . So a subset of Z np is P □ -definable ifand only if it is of the form {( a , . . . , a n ) ∈ Z np ∶ ( Exp ( pa ) , . . . , Exp ( pa n )) ∈ X } for a semialgebraic subset X of ( Z × p ) n . Proposition 11.6 shows that P □ defines an isomorphic copy of ( Q p , + , × ) . So if P ≺ N is highly saturated then N Sh interprets ( Q p , + , × ) , hence P is non-modular.We expect that N does not interpret an infinite field, but we do not have a proof. A p -adic completion conjecture.Conjecture 3. Suppose Z is a dp-minimal expansion of ( Z , + , Val p ) . Then thestructure induced on Z by Z □ is interdefinable with Z Sh and every Z Sh -definablesubset of Z n is of the form X ∩ Y where X is a Z □ -definable subset of Z np and Y isa ( Z , + ) -definable subset of Z n . The analogue of Conjecture 3 for dp-minimal expansions of divisible archimedeanordered groups is proven in [42]. We can prove a converse to Conjecture 3.
Proposition 11.7.
Let Y be an expansion of ( Z p , + , Val p ) and Z be the structureinduced on Z by Y . Suppose Y is dp-minimal and every Z -definable subset of Z n isof the form X ∩ Y where X is a Y -definable subset of Z np and Y is a ( Z , + ) -definablesubset of Z n . Then Z is dp-minimal.Proof. NIP formulas are closed under conjunctions so Z is NIP. So it suffices toshow that Z is inp-minimal. Inspection of the proof of Proposition 10.2 reveals thatour proof on inp-minimality for P only uses the following facts about ( Z p , + , × ) :(1) ( Z p , + , × ) is inp-minimal, and(2) every definable unary set in every elementary extension of Z p is the unionof a finite set and a definable open set.It follows from Fact 11.1 that any dp-minimal expansion of ( Z p , + , Val p ) satisfies ( ) . So the proof of Proposition 10.2 shows that Z is inp-minimal. (cid:3) p -adic elliptic curves? We give a conjectural construction of uncountably many dp-minimal expansionsof ( Z , + , Val p ) . Fix β ∈ p Z p . Then β Z is a closed subgroup of Q × p . It is a well-known theorem of Tate [44] that there is an elliptic curve E β defined over Q p and asurjective p -adic analytic group homomorphism ξ β ∶ Q × p → E β ( Q p ) with kernel β Z .Note that ξ β is injective on 1 + p Z p as ( + p Z p ) ∩ β Z = { } . We let χ β be the injective p -adic analytic homomorphism ( Z p , + ) → E β ( Q p ) given by χ β ( a ) ∶ = ξ β ( Exp ( pa )) , Y β be the structure induced on Z p by ( Q p , + , × ) and χ β , and E β be the structureinduced on Z by ( Q p , + , × ) and χ β . So E β is the structure induced on Z by Y β . Proposition 12.1. Y β expands ( Z p , + , Val p ) and E β expands ( Z , + , Val p ) . Proposition 12.1 requires some p -adic metric geometry. We letVal p ( a ) = min { Val p ( a ) , . . . , Val p ( a m )} for all a = ( a , . . . , a m ) ∈ Q mp . If X, Y are subsets of Q mp then f ∶ X → Y is an isometry if f is a bijection andVal p ( f ( a ) − f ( a ′ )) = Val p ( a − a ′ ) for all a, a ′ ∈ X. Suppose
X, Y are p -adic analytic submanifolds of Q mp . We let T a X be the tangentspace of X at a ∈ X . Given a p -adic analytic map f ∶ X → Y we let d f ( a ) ∶ T a X → T f ( a ) Y be the differential of f at a ∈ X . Fact 12.2.
Suppose f ∶ X → Y is a p -adic analytic map between p -adic analyticsubmanifolds X, Y of Q mp . Fix a ∈ X and set b ∶ = f ( a ) . Suppose that d f ( a ) is anisometry T a X → T b Y . Then there is an open neighbourhood U of p such that f ( U ) is open and f gives an isometry U → f ( U ) . See [13, Proposition 7.1] for a proof of Fact 12.2 when
X, Y are open subsets of Q mp . This generalizes to p -adic analytic submanifolds as any d -dimensional p -adicanalytic submanifold of Q mp is locally isometric to Q dp , see for example [17, 5.2](Halupczok only discusses smooth p -adic algebraic sets but everything goes throughfor p -adic analytic submanifolds).We now prove Proposition 12.1. Proof.
To simplify notion we drop the subscript “ β ”. It is enough to prove thefirst claim. It is easy to see that Y defines + . We need to show that the set of ( a, a ′ ) ∈ Z p such that Val p ( a ) ≤ Val p ( a ′ ) is definable in E . Note that if A is a finitesubset of E ( Q p ) and f ∶ E ( Q p ) \ A → Q mp is a semialgebraic injection then E isinterdefinable with the structure induced on Z p by ( Q p , + , × ) and f ◦ χ . So we canreplace E ( Q p ) and χ with f ( E ( Q p ) \ A ) and f ◦ χ .We consider E ( Q p ) as a subset of P ( Q p ) via the Weierstrass embedding. Let ι ∶ Q p → P ( Q p ) be the inclusion ι ( a, a ′ ) = [ a ∶ a ′ ∶ ] , U be the image of ι , and E ∶ = ι − ( E ( Q p )) . Recall that E ( Q p ) \ U is a singleton and E is a p -adic analyticsubmanifold of Q p . Let ζ ∶ Z p → E be ζ ∶ = ι − ◦ χ . So E is interdefinable with thestructure induced on Z p by ( Q p , + , × ) and ζ .Let e ∶ = ζ ( ) and identify T Z p with Q p . Note that d ζ ( ) is a bijection Q p → T e E .After making an affine change of coordinates if necessary we suppose d ζ ( ) is anisometry Q p → T e E . Applying Fact 12.2 we obtain n such that the restriction of ζ to p n Z p is an isometry onto its image. So for all a ∈ Z p we haveVal p ( ζ ( p n a ) − e ) = Val p ( p n a − ) = Val p ( a ) + n. So for all a, a ′ ∈ Z p we haveVal p ( a ) ≤ Val p ( a ′ ) if and only if Val p ( ζ ( p n a ) − e ) ≤ Val p ( ζ ( p n a ′ ) − e ) . Let X be the set of ( a, a ′ ) ∈ Z such that Val p ( ζ ( a ) − e ) ≤ Val p ( ζ ( a ′ ) − e ) , so X is definable in E . So for all ( a, a ′ ) ∈ Z p we have Val p ( a ) ≤ Val p ( a ′ ) if and only if ( p n a, p n a ′ ) ∈ X . So {( a, a ′ ) ∈ Z p ∶ Val p ( a ) ≤ Val p ( a ′ )} is definable in Y . (cid:3) We denote the group operation on E β ( Q p ) by ⊕ . Conjecture 4.
Suppose A is a finite rank subgroup of E β ( Q p ) . Then ( Q p , + , × , A ) is NIP , Th ( Q p , + , × ) is an open core of Th ( Q p , + , × , A ) , and every ( Q p , + , × , A ) -definable subset of A k is of the form X ∩ Y where X is an ( A, ⊕ ) -definable subsetof A k and Y is a semialgebraic subset of E β ( Q p ) k . Suppose Conjecture 4 holds. Under this assumption, Proposition 11.7 shows that E β is dp-minimal, an application of Proposition 11.5 shows that E □ β is interdefinablewith Y β , and an adaptation the proof of Theorem 9.4 shows that if E α and E β areinterdefinable then there is a semialgebraic group isomorphism E α ( Q p ) → E β ( Q p ) .So we obtain an uncountable collection of dp-minimal expansions of ( Z , + , Val p ) notwo of which are interdefinable. Conjecture 4 should hold for any one-dimensional p -adic semialgebraic group satis-fying a Mordell-Lang condition. One dimensional p -adic semialgebraic groups areclassified in [26].Suppose that H is a one-dimensional ( Q p , + , × ) -definable group. By [18] there is anopen subgroup V of H , a one-dimensional abelian algebraic group W defined over Q p , an open subgroup U of W ( Q p ) , and a ( Q p , + , × ) -definable group isomorphism V → U . So we suppose that H is W ( Q p ) , so in particular H is a p -adic analyticgroup. Let e be the identity of H and identify T e H with Q p . For sufficiently large n there is an open subgroup U of H and a p -adic analytic group isomorphismΞ ∶ ( p n Z p , + ) → U , this Ξ is the Lie-theoretic exponential, see [37, Corollary 19.9].Let H be the structure induced on Z by ( Q p , + , × ) and k ↦ Ξ ( p n k ) . It follows in thesame way as above that H expands ( Z , + , Val p ) . We expect that if H is semiabelianthen H is dp-minimal and H □ is interdefinable with the structure induced on Z p by ( Q p , + , × ) and a ↦ Ξ ( p n a ) .13. A general question
We briefly discuss the following question raised to us by Simon: Is there an abstractapproach to Z □ ? There are many ways in which one might try to make this moreprecise. For example: Given a sufficiently well behaved NIP structure M (per-haps dp-minimal, perhaps distal, perhaps expanding a group) can one construct acanonical structure M □ containing M such that M Sh is the structure induced on M by M □ , M □ is somehow “close to o-minimal”, and M □ is not too “big” relative to M ? In the completion of an NIP expansion H of an archimedean ordered abeliangroup, Fin is ⋁ -definable, Inf is ⋀ -definable, and the resulting logic topology on R agrees with the usual topology. The same thing happens for the other completionsdiscussed above. So perhaps there is a highly saturated M ≺ N , a set X which bothexternally definable and ⋁ -definable in N , an equivalence relation E on X whichis both externally definable and ⋀ -definable in N , such that M □ is the structureinduced on X / E by N Sh .The completions defined above are not always the “right” notion. Let P be the set ofprimes and fix q ∈ P . Consider ( Z , + , ( Val p ) p ∈ P ) as an expansion of ( Z , + , Val q ) ,one can show that ( Z , + , ( Val p ) p ∈ P ) □ is interdefinable with ( Z q , + , Val q ) . How-ever the “right” completion of ( Z , + , ( Val p ) p ∈ P ) is ( ̂ Z , + , ( ◁ p ) p ∈ P ) where ( ̂ Z , + ) is the profinite completion ∏ p ∈ P ( Z p , + ) of ( Z , + ) and we have a ◁ p b if andonly if Val p ( π p ( a )) < Val p ( π p ( b )) , where π p is the projection ̂ Z → Z p . Like-wise, if I is a Z -linearly independent subset of R \ Q then the completion of ( Z , + , ( C α ) α ∈ I ) should be the torus (( R / Z ) I , + , S α ) where we have S α ( a, b, c ) ifand only if C ( π α ( a ) , π α ( b ) , π α ( c )) where π α is the projection ( R / Z ) I → R / Z ontothe α th coordinate.When M expands a group it is tempting to try to define M □ via general ideas fromNIP group theory. Fix irrational α ∈ R / Z , let Z be a dp-minimal expansion of ( Z , + , C α ) , and Z ≺ N be highly saturated. One can show that N / N is isomor-phic as a topological group to R / Z and it seems likely that Z □ is interdefinablewith the structure on R / Z with an n -ary relation defining the image of X ∩ ( N ) n under the quotient map ( N ) n → ( R / Z ) n , for each N -definable X ⊆ N n . But thisbreaks in the p -adic setting, if ( Z , + , Val p ) ≺ N is highly saturated then N = N .Is there some general class of “complete” structures for which M and M □ are in-terdefinable? Suppose H is a dense subgroup of ( R , + ) and H is an NIP expansionof ( H, + , < ) . Then H and H □ are interdefinable if and only if H = R and H is in-terdefinable with the open core of H Sh . The Marker-Steinhorn theorem shows thatthese conditions are satisfied when H is an o-minimal expansion of ( R , + , < ) . If H is o-minimal then H □ is (up to interdefinibility) the unique elementary extension of H which expands ( R , + , < ) [49, 13.2.1]. (Laskowski and Steinhorn [24] showed thatthere is such an extension.) If H is weakly o-minimal then H □ is an elementaryexpansion of H if and only if H is o-minimal. Should M be “complete” if M □ is(up to interdefinability) an elementary extension of M ? If Z is a dp-minimal ex-pansion of ( Z p , + , Val p ) then must Z □ and Z be interdefinable, i.e. is there a p -adicMarker-Steinhorn generalizing Fact 3.7? References [1] E. Alouf. On dp-minimal expansions of the integers. arXiv:2001.11480 , 2020.[2] E. Alouf and C. d’Elb´ee. A new dp-minimal expansion of the integers.
J. Symb. Log. ,84(2):632–663, 2019.[3] M. Aschenbrenner, A. Dolich, D. Haskell, D. Macpherson, and S. Starchenko. Vapnik-Chervonenkis density in some theories without the independence property, II.
Notre DameJ. Form. Log. , 54(3-4):311–363, 2013.[4] M. Aschenbrenner, A. Dolich, D. Haskell, D. Macpherson, and S. Starchenko. Vapnik-Chervonenkis density in some theories without the independence property, I.
Trans. Amer.Math. Soc. , 368(8):5889–5949, 2016.[5] M. Aschenbrenner and L. van den Dries. Closed asymptotic couples.
J. Algebra , 225(1):309–358, 2000.[6] G. Boxall and P. Hieronymi. Expansions which introduce no new open sets.
J. Symbolic Logic ,77(1):111–121, 2012.[7] A. Chernikov and P. Simon. Externally definable sets and dependent pairs.
Israel J. Math. ,194(1):409–425, 2013.[8] A. Chernikov and P. Simon. Externally definable sets and dependent pairs II.
Trans. Amer.Math. Soc. , 367(7):5217–5235, 2015.[9] G. Conant. There are no intermediate structures between the group of integers and presburgerarithmetic.
J. Symb. Log. , 83(1):187–207, 2018.[10] G. Conant and A. Pillay. Stable groups and expansions of ( Z , + , ) . Fundamenta Mathemat-icae , to appear, 2016.[11] F. Delon. D´efinissabilit´e avec param`etres ext´erieurs dans Q p et R . Proc. Amer. Math. Soc. ,106(1):193–198, 1989.[12] A. Dolich and J. Goodrick. Strong theories of ordered Abelian groups.
Fund. Math. ,236(3):269–296, 2017.[13] H. Gl¨ockner. Implicit functions from topological vector spaces to banach spaces.
Israel Journalof Mathematics , 155(1):205–252, Dec. 2006.[14] A. B. Gorman, P. Hieronymi, and E. Kaplan. Pairs of theories satisfying a mordell-langcondition, 2018.[15] F. Guignot.
Th´eorie des mod`eles des groupes ab´eliens valu´es . PhD thesis, Phd thesis, Paris,2016.[16] A. G¨unaydı n and P. Hieronymi. The real field with the rational points of an elliptic curve.
Fund. Math. , 211(1):15–40, 2011.[17] I. Halupczok. Trees of definable sets in Z p . In R. Cluckers, J. Nicaise, and J. Sebag, editors, Motivic Integration and its Interactions with Model Theory and Non-Archimedean Geometry ,pages 87–107. Cambridge University Press. [18] E. Hrushovski and A. Pillay. Groups definable in local fields and pseudo-finite fields.
IsraelJ. Math. , 85(1–3):203–262, 1994.[19] F. Jahnke, P. Simon, and E. Walsberg. Dp-minimal valued fields.
J. Symb. Log. , 82(1):151–165, 2017.[20] W. Johnson.
Fun with fields . PhD thesis, 2016.[21] W. Johnson. The canonical topology on dp-minimal fields.
J. Math. Log. , 18(2):1850007, 23,2018.[22] A. S. Kechris.
Classical descriptive set theory , volume 156 of
Graduate Texts in Mathematics .Springer-Verlag, New York, 1995.[23] F.-V. Kuhlmann. Abelian groups with contractions. II. Weak o-minimality. In
Abelian groupsand modules (Padova, 1994) , volume 343 of
Math. Appl. , pages 323–342. Kluwer Acad. Publ.,Dordrecht, 1995.[24] M. C. Laskowski and C. Steinhorn. On o-minimal expansions of Archimedean ordered groups.
J. Symbolic Logic , 60(3):817–831, 1995.[25] O. Le Gal. A generic condition implying o-minimality for restricted C ∞ -functions. Ann. Fac.Sci. Toulouse Math. (6) , 19(3-4):479–492, 2010.[26] J. P. A. Lpez. One dimensional groups definable in the p-adic numbers, 2018,arxiv:1811.09854.[27] J. J. Madden and C. M. Stanton. One-dimensional Nash groups.
Pacific J. Math. , 154(2):331–344, 1992.[28] N. Mariaule. The field of p -adic numbers with a predicate for the powers of an integer. J.Symb. Log. , 82(1):166–182, 2017.[29] N. Mariaule. Model theory of the field of p -adic numbers expanded by a multiplicative sub-group. arXiv:1803.10564 , 2018.[30] D. Marker and C. I. Steinhorn. Definable types in o-minimal theories. J. Symbolic Logic ,59(1):185–198, 1994.[31] B. Mazur. Abelian varieties and the Mordell-Lang conjecture. In
Model theory, algebra, andgeometry , volume 39 of
Math. Sci. Res. Inst. Publ. , pages 199–227. Cambridge Univ. Press,Cambridge, 2000.[32] C. Michaux and R. Villemaire. Presburger arithmetic and recognizability of sets of naturalnumbers by automata: new proofs of Cobham’s and Semenov’s theorems.
Ann. Pure Appl.Logic , 77(3):251–277, 1996.[33] A. Onshuus and A. Usvyatsov. On dp-minimality, strong dependence and weight.
J. SymbolicLogic , 76(3):737–758, 2011.[34] M. Otero, Y. Peterzil, and A. Pillay. On groups and rings definable in o-minimal expansionsof real closed fields.
Bull. London Math. Soc. , 28(1):7–14, 1996.[35] D. Palac´ın and R. Sklinos. On superstable expansions of free abelian groups.
Notre DameJournal of Formal Logic , 59(2):157–169, 2018.[36] A. Pillay. On groups and fields definable in o -minimal structures. J. Pure Appl. Algebra ,53(3):239–255, 1988.[37] P. Schneider. p-Adic Lie Groups . Springer Berlin Heidelberg, 2011.[38] S. Shelah. Dependent first order theories, continued.
Israel J. Math. , 173:1–60, 2009.[39] M. Shiota. O-minimal Hauptvermutung for polyhedra I.
Invent. Math. , 196(1):163–232, 2014.[40] P. Simon. On dp-minimal ordered structures.
J. Symbolic Logic , 76(2):448–460, 2011.[41] P. Simon.
A guide to NIP theories , volume 44 of
Lecture Notes in Logic . Cambridge UniversityPress, 2015.[42] P. Simon and E. Walsberg. Dp and other minimalities.
Preprint , arXiv:1909.05399, 2019.[43] P. Simon and E. Walsberg. Tame topology over dp-minimal structures.
Notre Dame J. Form.Log. , 60(1):61–76, 2019.[44] J. Tate. A review of non-Archimedean elliptic functions. In
Elliptic curves, modular forms,& Fermat’s last theorem (Hong Kong, 1993) , Ser. Number Theory, I, pages 162–184. Int.Press, Cambridge, MA, 1995.[45] M. C. Tran and E. Walsberg. A family of dp-minimal expansions of ( Z ; + ) , 2017.[46] M. A. Tychonievich. Tameness results for expansions of the real field by groups . ProQuestLLC, Ann Arbor, MI, 2013. Thesis (Ph.D.)–The Ohio State University.[47] L. van den Dries.
Tame topology and o-minimal structures , volume 248 of
London Mathe-matical Society Lecture Note Series . Cambridge University Press, Cambridge, 1998. [48] E. Walsberg. An nip structure which does not interpret an infinite group but whose shelahexpansion interprets an infinite field. arXiv:1910.13504 , 2019.[49] E. Walsberg. Externally definable quotients and nip expansions of the real ordered additivegroup, 2019, arXiv:1910.10572.[50] V. Weispfenning. Elimination of quantifiers for certain ordered and lattice-ordered abeliangroups. Bull. Soc. Math. Belg. S´er. B , 33(1):131–155, 1981.
Department of Mathematics, Statistics, and Computer Science, Department of Math-ematics, University of California, Irvine, 340 Rowland Hall (Bldg.
E-mail address : [email protected] URL ::