Dp-minimal expansions of $(\mathbb{Z},+)$ via dense pairs via Mordell-Lang

We construct new dp-minimal expansions of $(\mathbb{Z},+)$ and take some steps towards classifying dp-minimal expansions of $(\mathbb{Z},+)$ which define either a dense cyclic group order or a $p$-adic valuation. (Every known proper dp-minimal expansion of $(\mathbb{Z},+)$ 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 strong form of $\mathrm{NIP}$ which is broad enough to include many interesting structures and narrow enough to have very strong consequences. O-minimality and related notions imply dp-minimality. Johnson [20] classified dp-minimal fields. Simon [40] showed that an expansion of $(\mathbb{R},+,<)$ is dp-minimal if and only if it is o-minimal. We summarize recent work on dp-minimal expansions of $(\mathbb{Z},+)$ in Section 6.

It was an open question for some years whether every proper dp-minimal expansion of $(\mathbb{Z},+)$ is interdefinable with $(\mathbb{Z},+,<)$ [3, Question 5.32]. It turns out that this question was essentially answered before it was posed, in work on “dense pairs”. We will show, applying work of Hieronymi and Günaydin [16], that if $\mathbb{S}$ is the unit circle, $t\in\mathbb{R}$ is irrational, and $\chi:\mathbb{Z}\to\mathbb{S}$ is the character $\chi(k):=e^{2\pi itk}$ then the structure induced on $\mathbb{Z}$ by $(\mathbb{R},+,\times)$ and $\chi$ is dp-minimal.

Indeed, for every known dp-minimal expansion $\mathscr{Z}$ of $(\mathbb{Z},+)$ there is a dp-minimal field $\mathscr{K}$, a semiabelian $\mathscr{K}$-variety $V$, and a character $\chi:\mathbb{Z}\to V(\mathscr{K})$ such that the structure $\mathscr{Z}_{\chi}$ induced on $\mathbb{Z}$ by $\mathscr{K}$ and $\chi$ is dp-minimal and $\mathscr{Z}$ is a reduct of $\mathscr{Z}_{\chi}$.

We now briefly describe how the known dp-minimal expansions of $(\mathbb{Z},+)$ fall into this framework. It follows directly from the Mordell-Lang conjecture that if $\beta\in\mathbb{C}^{\times}$ is not a root of unity then the structure induced on $\mathbb{Z}$ by $(\mathbb{C},+,\times)$ and the character $k\mapsto\beta^{k}$ is interdefinable with $(\mathbb{Z},+)$. It follows from a result of Tychonievich [46, Theorem 4.1.2] that if $\beta\in\mathbb{R}^{\times}\setminus\{-1,1\}$ then the structure induced on $\mathbb{Z}$ by $(\mathbb{R},+,\times)$ and the character $k\mapsto\beta^{k}$ is interdefinable with $(\mathbb{Z},+,<)$. (It is also shown in [29] that if $\beta\in\mathbb{Q}^{\times}_{p}$ and $\operatorname{Val_{p}}(\beta)\neq 0$ then $(\mathbb{Z},+,<)$ is interdefinable with the structure induced on $\mathbb{Z}$ by $(\mathbb{Q}_{p},+,\times)$ and $k\mapsto\beta^{k}$.) Below we apply work of Mariaule [29] to show that there is $\beta\in 1+p\mathbb{Z}_{p}$ such that the structure induced on $\mathbb{Z}$ by $(\mathbb{Q}_{p},+,\times)$ and $k\mapsto\beta^{k}$ is a dp-minimal expansion of $(\mathbb{Z},+,\operatorname{Val_{p}})$. The only other previously known dp-minimal expansion of $(\mathbb{Z},+)$ is $(\mathbb{Z},+,S)$ where $S$ is a dense cyclic group order [45]. There is a unique $\beta\in\mathbb{S}$ such that $S$ is the pullback of the clockwise cyclic order on $\mathbb{S}$ by $k\mapsto\beta^{k}$. So the structure induced on $\mathbb{Z}$ by $(\mathbb{R},+,\times)$ and $k\mapsto\beta^{k}$ is a dp-minimal expansion of $(\mathbb{Z},+,S)$.

We produce uncountably many new dp-minimal expansions of $(\mathbb{Z},+,S)$. Let $\mathbb{E}$ be an elliptic curve defined over $\mathbb{R}$, $\mathbb{E}^{0}(\mathbb{R})$ be the connected component of the identity, and $\chi:\mathbb{Z}\to\mathbb{E}^{0}(\mathbb{R})$ be a character such that $S$ is the pullback by $\chi$ of the natural cyclic order on $\mathbb{E}^{0}(\mathbb{R})$. We apply [16] to show that the structure $\mathscr{Z}_{\mathbb{E}}$ induced on $\mathbb{Z}$ by $(\mathbb{R},+,\times)$ and $\chi$ is a proper dp-minimal expansion of $(\mathbb{Z},+,S)$. We also show that $\mathbb{E}^{0}(\mathbb{R})$ may be recovered up to semialgebraic isomorphism from $\mathscr{Z}_{\mathbb{E}}$. It follows that there is an uncountable family of dp-minimal expansions of $(\mathbb{Z},+,S)$ no two of which are interdefinable.

We describe how $\mathbb{E}^{0}(\mathbb{R})$ may be recovered from $\mathscr{Z}_{\mathbb{E}}$. Let $C$ be the usual clockwise cyclic order on $\mathbb{R}/\mathbb{Z}$. Given any dp-minimal expansion $\mathscr{Z}$ of $(\mathbb{Z},+,S)$ we define a completion $\mathscr{Z}^{\square}$ of $\mathscr{Z}$, this $\mathscr{Z}^{\square}$ is an o-minimal expansion of $(\mathbb{R}/\mathbb{Z},+,C)$ canonically associated to $\mathscr{Z}$. We show that $\mathscr{Z}^{\square}_{\mathbb{E}}$ is the structure induced on $\mathbb{R}/\mathbb{Z}$ by $(\mathbb{R},+,\times)$ and the unique (up to sign) topological group isomorphism $\mathbb{R}/\mathbb{Z}\to\mathbb{E}^{0}(\mathbb{R})$. The recovery of $\mathbb{E}^{0}(\mathbb{R})$ from $\mathscr{Z}^{\square}_{\mathbb{E}}$ is a special case of a canonical correspondence between

1. (1)

   non-modular o-minimal expansions $\mathscr{C}$ of $(\mathbb{R}/\mathbb{Z},+,C)$, and

2. (2)

   pairs $\langle\mathscr{R},\mathbb{H}\rangle$ where $\mathscr{R}$ is an o-minimal expansion of $\mathscr{R}$ and $\mathbb{H}$ is an $\mathscr{R}$-definable circle group.

Given $\langle\mathscr{R},\mathbb{H}\rangle$, $\mathscr{C}$ is unique up to interdefinibility. Given $\mathscr{C}$, $\mathscr{R}$ is unique up to interdefinibility and $\mathbb{H}$ is unique up to $\mathscr{R}$-definable isomorphism.

We describe $\mathscr{Z}^{\square}$ for a fixed dp-minimal expansion $\mathscr{Z}$ of $(\mathbb{Z},+,S)$. Let $\psi:\mathbb{Z}\to\mathbb{R}/\mathbb{Z}$ be the unique character such that $S$ is the pullback of $C$ by $\psi$. Let $\mathscr{Z}\prec\mathscr{N}$ be highly saturated, $\mathscr{N}^{\mathrm{Sh}}$ be the Shelah expansion of $\mathscr{N}$, and $\mathbf{Inf}$ be the natural subgroup of infinitesimals in $\mathscr{N}$. We identify $N/\mathbf{Inf}$ with $\mathbb{R}/\mathbb{Z}$ and identify the quotient map $N\to\mathbb{R}/\mathbb{Z}$ with the standard part map. As $\mathbf{Inf}$ is $\mathscr{N}^{\mathrm{Sh}}$-definable we regard $\mathbb{R}/\mathbb{Z}$ as an imaginary sort of $\mathscr{N}^{\mathrm{Sh}}$. A slight adaptation of [49] shows that the following structures are interdefinable:

1. (1)

   The structure on $\mathbb{R}/\mathbb{Z}$ with an $n$-ary relation defining the closure in $(\mathbb{R}/\mathbb{Z})^{n}$ of $\{(\psi(a_{1}),\ldots,\psi(a_{n})):(a_{1},\ldots,a_{n})\in X\}$ for each $\mathscr{Z}^{\mathrm{Sh}}$-definable $X\subseteq\mathbb{Z}^{n}$,

2. (2)

   The structure on $\mathbb{R}/\mathbb{Z}$ with an $n$-ary relation defining the image under the standard part map $N^{n}\to(\mathbb{R}/\mathbb{Z})^{n}$ of each $\mathscr{N}$-definable subset of $N^{n}$,

3. (3)

   The structure induced on $\mathbb{R}/\mathbb{Z}$ by $\mathscr{N}^{\mathrm{Sh}}$.

We refer to any of these structure as $\mathscr{Z}^{\square}$. It follows from $(3)$ that $\mathscr{Z}^{\square}$ is dp-minimal, a slight adaptation of [40] shows that any dp-minimal expansion of $(\mathbb{R}/\mathbb{Z},+,C)$ is o-minimal, so $\mathscr{Z}^{\square}$ is o-minimal. We will see that the structure induced on $\mathbb{Z}$ by $\mathscr{Z}^{\square}$ and $\psi$ is a reduct of the Shelah expansion of $\mathscr{Z}$. In future work we intend to show that these two are interdefinable. This will reduce the question “what are the dp-minimal expansions of $(\mathbb{Z},+,S)$ to “for which o-minimal expansions $\mathscr{C}$ of $(\mathbb{R}/\mathbb{Z},+,C)$ is the structure induced on $\mathbb{Z}$ by $\mathscr{C}$ and $\psi$ dp-minimal”?

We also define an analogous completion $\mathscr{P}^{\square}$ of a dp-minimal expansion $\mathscr{P}$ of $(\mathbb{Z},+,\operatorname{Val_{p}})$, this $\mathscr{P}^{\square}$ is a dp-minimal expansion of $(\mathbb{Z}_{p},+,\operatorname{Val_{p}})$. The structure induced on $\mathbb{Z}$ by $\mathscr{P}^{\square}$ is reduct of $\mathscr{P}^{\mathrm{Sh}}$. We expect the induced structure to be interdefinable with $\mathscr{P}^{\mathrm{Sh}}$.

It is easy to see that $\mathscr{Z}^{\square}_{\mathbb{E}}$ defines an isomorphic copy of $(\mathbb{R},+,\times)$. It follows that if $\mathscr{Z}_{\mathbb{E}}\prec\mathscr{N}_{\mathbb{E}}$ is highly saturated then the Shelah expansion of $\mathscr{N}_{\mathbb{E}}$ interprets $(\mathbb{R},+,\times)$, so $\mathscr{Z}_{\mathbb{E}}$ should be “non-modular”. (One can show that $\mathscr{Z}_{\mathbb{E}}$ itself does not interpret an infinite field.) At present there is no published notion of modularity for general $\mathrm{NIP}$ structures, but there should be a notion of modularity for $\mathrm{NIP}$ (or possibly just distal) structures which satisfies the following.

1. (A1)

   A modular structure cannot interpret an infinite field.

2. (A2)

   Abelian groups, linearly (or cyclically) ordered abelian groups, $\mathrm{NIP}$ valued abelian groups, and ordered vector spaces are modular.

3. (A3)

   If $\mathscr{M}$ is modular and the structure induced on $A\subseteq M^{n}$ by $\mathscr{M}$ eliminates quantifiers then the induced structure is modular. In particular the Shelah expansion of a modular structure is modular. (Recall that the induced structure eliminates quantifiers if and only if every definable subset of $A^{m}$ is of the form $A^{m}\cap X$ for $\mathscr{M}$-definable $X$.)

4. (A4)

   An o-minimal structure is modular if and only if it does not define an infinite field. (This should follow from the Peterzil-Starchenko trichotomy.)

In this paper we will assume that there is a notion of modularity satisfying these conditions, but none of our results fail if this is not true. $($A$2)$ implies that all previously known dp-minimal expansions of $(\mathbb{Z},+)$ are modular. $($A$1)$ and $($A$3)$ imply that if $\mathscr{Z}^{\square}$ defines $(\mathbb{R},+,\times)$ then $\mathscr{Z}$ is non-modular. If $\mathscr{Z}^{\square}$ does not define $(\mathbb{R},+,\times)$ then $($A$4)$ implies that $\mathscr{Z}^{\square}$ is modular. We expect that if $\mathscr{Z}^{\square}$ is modular then $\mathscr{Z}$ is modular.

We will see that if $\mathscr{P}$ is the structure induced on $\mathbb{Z}$ by $(\mathbb{Q}_{p},+,\times)$ and the character $k\mapsto\operatorname{Exp}(pk)$ then $\mathscr{P}^{\square}$ is interdefinable with the structure induced on $\mathbb{Z}_{p}$ by $(\mathbb{Q}_{p},+,\times)$ and the isomorphism $(\mathbb{Z}_{p},+)\to(1+p\mathbb{Z}_{p},\times),a\mapsto\operatorname{Exp}(pa)$. It follows that the Shelah expansion of a highly saturated $\mathscr{P}\prec\mathscr{N}$ interprets $(\mathbb{Q}_{p},+,\times)$, so $\mathscr{P}$ is non-modular. We again expect that $\mathscr{P}$ is modular if and only if $\mathscr{P}^{\square}$ is modular, but we do not have a modular/non-modular dichotomy for dp-minimal expansions of $(\mathbb{Z}_{p},+,\operatorname{Val_{p}})$ (we lack a $p$-adic Peterzil-Starchenko.) It seems reasonable to conjecture that a dp-minimal expansion of $(\mathbb{Z}_{p},+,\operatorname{Val_{p}})$ is non-modular if and only if it defines an isomorphic copy of $(\mathbb{Q}_{p},+,\times)$.

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 groups, and in Section 5 we recall some basic facts on definable groups in o-minimal expansions of $(\mathbb{R},+,\times)$. In Section 6 we survey previous work on dp-minimal expansions of $(\mathbb{Z},+)$. In Section 7 we construct new dp-minimal expansions of $(\mathbb{Z},+,S)$ where $S$ is a dense cyclic group order. In Section 8 we describe the o-minimal completion of a strongly dependent expansion of $(\mathbb{Z},+,S)$. We also show that the Shelah expansion $(\mathbb{Z},+,S)^{\mathrm{Sh}}$ of $(\mathbb{Z},+,S)$ is interdefinable with the structure induced on $\mathbb{Z}$ by $(\mathbb{R}/\mathbb{Z},+,C)$ and $\psi$, where $\psi:\mathbb{Z}\to\mathbb{R}/\mathbb{Z}$ is the unique character such that $S$ is the pullback of $C$ by $\psi$. It follows that $(\mathbb{Z},+,S)^{\mathrm{Sh}}$ is a reduct of each of our dp-minimal expansions of $(\mathbb{Z},+,S)$. In Section 9 we show that two of our dp-minimal expansions of $(\mathbb{Z},+,S)$ are interdefinable if and only if the associated semialgebraic circle groups are semialgebraically isomorphic. In Section 10 we construct a new dp-minimal expansion $\mathscr{P}$ of $(\mathbb{Z},+,\operatorname{Val_{p}})$ and in Section 11 we describe the $p$-adic completion of a dp-minimal expansion of $(\mathbb{Z},+,\operatorname{Val_{p}})$. In Section 12 we give a conjecture which implies that one can construct uncountably many dp-minimal expansions of $(\mathbb{Z},+,\operatorname{Val_{p}})$ from $p$-adic elliptic curves. Finally, in Section 13 we briefly discuss the question of whether our completion constructions are special cases of an abstract model-theoretic completion.

Given a tuple $x=(x_{1},\ldots,x_{n})$ of variables we let $|x|=n$. Throughout $n$ is a natural number, $m,k,l$ are integers, $t,r,\lambda,\eta$ are real numbers, and $\alpha$ is an element of $\mathbb{R}/\mathbb{Z}$. Suppose $\alpha\in\mathbb{R}/\mathbb{Z}$. We let $\psi_{\alpha}$ denote the character $\mathbb{Z}\to\mathbb{R}/\mathbb{Z}$ given by $\psi_{\alpha}(k)=\alpha k$. We say that $\alpha$ is irrational if $\alpha=s+\mathbb{Z}$ for $s\in\mathbb{R}\setminus\mathbb{Q}$. Note that $\alpha$ is irrational if and only if $\psi_{\alpha}$ is injective.

All structures are first order and “definable” means “first-order definable, possibly with parameters”. Suppose $\mathscr{M}$, $\mathscr{N}$, and $\mathscr{O}$ are structures on a common domain $M$. Then $\mathscr{M}$ is a reduct of $\mathscr{O}$ (and $\mathscr{O}$ is an expansion of $\mathscr{M}$) if every $\mathscr{M}$-definable subset of every $M^{n}$ is $\mathscr{O}$-definable, $\mathscr{M}$ and $\mathscr{O}$ are interdefinable if each is a reduct of the other, $\mathscr{M}$ is a proper reduct of $\mathscr{O}$ (and $\mathscr{O}$ is a proper expansion of $\mathscr{M}$) if $\mathscr{M}$ is a reduct of $\mathscr{O}$ and $\mathscr{M}$ is not interdefinable with $\mathscr{O}$, and $\mathscr{N}$ is intermediate between $\mathscr{M}$ and $\mathscr{O}$ if $\mathscr{M}$ is a proper reduct of $\mathscr{N}$ and $\mathscr{N}$ is a proper reduct of $\mathscr{O}$.

Given a set $A$ and an injection $f:A\to M^{m}$ we say that the structure induced on $A$ by $\mathscr{M}$ and $f$ is the structure on $A$ with an $n$-ary relation defining $\{(a_{1},\ldots,a_{n})\in A^{n}:((f(a_{1})\ldots,f(a_{n}))\in Y\}$ for every $\mathscr{M}$-definable $Y\subseteq M^{nm}$. If $A$ is a subset of $M^{m}$ and $f:A\to M^{m}$ is the identity we refer to this as the structure induced on $A$ by $\mathscr{M}$.

We let $\operatorname{Cl}(X)$ denote the closure of a subset $X$ of a topological space.

Suppose $L\subseteq L^{\prime}$ are languages containing $<$, $\mathscr{R}^{\prime}$ is an $L^{\prime}$-structure expanding $(\mathbb{R},<)$ and $\mathscr{R}$ is the $L$-reduct of $\mathscr{R}^{\prime}$. The open core of $\mathscr{R}^{\prime}$ is the reduct of $\mathscr{R}^{\prime}$ generated by all closed $\mathscr{R}^{\prime}$-definable sets. Furthermore $\mathrm{Th}(\mathscr{R})$ is an open core of $\mathrm{Th}(\mathscr{R}^{\prime})$ if, whenever $\mathscr{R}^{\prime}\prec\mathscr{N}^{\prime}$ then the $L^{\prime}$-reduct of $\mathscr{N}^{\prime}$ is interdefinable with the open core of $\mathscr{N}^{\prime}$. This notion clearly makes sense in much broader generality.

We use “semialgebraic” as a synonym of either “$(\mathbb{R},+,\times)$-definable” or “$(\mathbb{Q}_{p},+,\times)$-definable”. It will be clear in context which we mean.

Let $\mathscr{M}$ be a structure and $\mathscr{M}\prec\mathscr{N}$ be highly saturated.

We give basic definitions and results concerning cyclically ordered groups. We also set 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 $a,b,c\in G$,

1. (1)

   if $S(a,b,c)$, then $S(b,c,a)$,

2. (2)

   if $S(a,b,c)$, then $\neg S(c,b,a)$,

3. (3)

   if $S(a,b,c)$ and $S(a,c,d)$ then $S(a,b,d)$,

4. (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 $\{b\in G:S(a,b,c)\}$ for some $a,c\in G$, likewise define closed and half open intervals. A subset of $G$ is $S$-convex if it is the union 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 $+$-invariant cyclic order. Suppose $S$ is a cyclic group order on $(G,+)$. A subset of $G$ is an $\mathbf{S}$ tmc set if it is of the form $a+mJ$ for $S$-convex $J\subseteq G$ and $a\in G$. We drop the “$S$” when it is clear from context.

Note that $\{(a,b,c)\in G^{2}:S(c,b,a)\}$ is a cyclic group order which we refer to as the opposite of $S$. (If $<$ is a linear group order on $(G,+)$ then $\{(a,b)\in G^{2}:b<a\}$ is also a linear group order which we refer to as the opposite of $<$.)

Throughout $C$ is the cyclic group order on $(\mathbb{R}/\mathbb{Z},+)$ such that whenever $t,t^{\prime},t^{\prime\prime}\in\mathbb{R}$ and $0\leq t,t^{\prime},t^{\prime\prime}<1$ then $C(t+\mathbb{Z},t^{\prime}+\mathbb{Z},t^{\prime\prime}+\mathbb{Z})$ holds if and only if either $t<t^{\prime}<t^{\prime\prime}$, $t^{\prime}<t^{\prime\prime}<t$, or $t^{\prime\prime}<t<t^{\prime}$. Given irrational $\alpha\in\mathbb{R}/\mathbb{Z}$ we let $\boldsymbol{C_{\alpha}}$ be the cyclic group order on $(\mathbb{Z},+)$ where $C_{\alpha}(k,k^{\prime},k^{\prime\prime})$ if and only if $C(\alpha k,\alpha k^{\prime},\alpha k^{\prime\prime})$, so $C_{\alpha}$ is the pullback of $C$ by $\psi_{\alpha}$. Every dense cyclic group order on $(\mathbb{Z},+)$ is of this form for unique $\alpha\in\mathbb{R}/\mathbb{Z}$.

Let $<$ be a linear group order on $(G,+)$. There are two associated cyclic orders:

$$S_{<}:=\{(a,b,c)\in G^{3}:(a<b<c)\vee(b<c<a)\vee(c<a<b)\},$$

and

$$S_{>}:=\{(a,b,c)\in G^{3}:(c<b<a)\vee(b<a<c)\vee(a<c<b)\}.$$

Note that $S_{<}$ is the opposite of $S_{>}$. See for example [45] for a proof of Fact 4.1.

We will frequently apply Fact 4.2, which is elementary and left to the reader.

We recall some basic facts from the extensive theory of definable groups in o-minimal structures. Throughout this section $\mathscr{R}$ is an o-minimal expansion of $(\mathbb{R},+,\times)$, $\mathbb{H}$ is an $\mathscr{R}$-definable group, and “dimension” without modification is the o-minimal dimension.

Fact 5.1 follows from work of Pillay [36] and [47, 10.1.8].

We let $\mathcal{T}_{\mathbb{H}}$ be the canonical group topology on $\mathbb{H}$ and consider $\mathbb{H}$ as a topological group. Recall that any connected topological group of topological dimension one is isomorphic (as a topological group) to either $(\mathbb{R},+)$ or $(\mathbb{R}/\mathbb{Z},+)$. It follows that if $\mathbb{H}$ is one-dimensional and connected then $\mathbb{H}$ is isomorphic as a topological group to either $(\mathbb{R},+)$ or $(\mathbb{R}/\mathbb{Z},+)$. In the first case we say that $\mathbb{H}$ is a line group, in the second case $\mathbb{H}$ is a circle group.

Suppose $X$ is an $\mathscr{R}$-definable subset of $\mathbb{R}^{m}$. An easy application of the good directions lemma [47, Theorem 4.2] shows that if $X$ is homeomorphic to $\mathbb{R}$ then there is an $\mathscr{R}$-definable homeomorphism $X\to\mathbb{R}$ and if $X$ is homeomorphic to $\mathbb{R}/\mathbb{Z}$ then there is an $\mathscr{R}$-definable homeomorphism from $X$ to the unit circle. (The analogous fact fails in higher dimensions, there are homeomorphic semialgebraic sets $X,X^{\prime}$ for which there is no homeomorphism $X\to X^{\prime}$ definable in an o-minimal expansion of $(\mathbb{R},+,\times)$, this is a consequence of Shiota’s o-minimal Hauptvermutung [39] together with the failure of the classical Hauptvermutung.) Fact 5.2 easily follows.

So if $\mathbb{H}$ is one-dimensional and connected and $A$ is a subgroup of $\mathbb{H}$ then we may speak without ambiguity of a tmc subset of $A$.

Finally we recall the interpretation-rigidity theorem for o-minimal expansions of $(\mathbb{R},+,\times)$. Fact 5.3 is due to Otero, Peterzil, and Pillay [34].

We survey what is known about dp-minimal expansions of $(\mathbb{Z},+)$.

The first result on dp-minimal expansions of $(\mathbb{Z},+)$ is Fact 6.1, proven in [4, Proposition 6.6]. Fact 6.1 follows easily from two results, the Michaux-Villemaire theorem [32] that there are no proper $(\mathbb{Z},+,<)$-minimal expansions of $(\mathbb{Z},+,<)$, and Simon’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.

The authors of [4] raised the question of whether there is a dp-minimal expansion of $(\mathbb{Z},+)$ which is not a reduct of $(\mathbb{Z},+,<)$. Conant and Pillay [10] proved Fact 6.2. Their proof relies on earlier work of Palacín and Sklinos [35], who apply the Buechler dichotomy theorem and other sophisticated tools of stability theory.

Conant [9] proved Fact 6.3 via a geometric analysis of $(\mathbb{Z},+,<)$-definable sets. Facts 6.2 and 3.8 show that a proper dp-minimal expansion of $(\mathbb{Z},+)$ is not $\mathrm{Th}(\mathbb{Z},+)$-minimal. Alouf and d’Elbée [2] used this to give a quicker proof of Fact 6.3.

Alouf and d’Elbée [2] proved Fact 6.4. Given a prime $p$ we let $\operatorname{Val_{p}}$ be the $p$-adic valuation on $(\mathbb{Z},+)$ and $\prec_{p}$ be the partial order on $\mathbb{Z}$ where $m\prec_{p}n$ if and only if $\operatorname{Val_{p}}(m)<\operatorname{Val_{p}}(n)$. We can view $(\mathbb{Z},+,\operatorname{Val_{p}})$ as either $(\mathbb{Z},+,\prec_{p})$ or as the two sorted structure with disjoint sorts $\mathbb{Z}$ and $\mathbb{N}\cup\{\infty\}$, addition on $\mathbb{Z}$, and $\operatorname{Val_{p}}:\mathbb{Z}\to\mathbb{N}\cup\{\infty\}$. It makes no difference which of these two options we take.

Alouf and d’Elbée also show that $(\mathbb{Z},+,(\operatorname{Val_{p}})_{p\in I})$ has dp-rank $|I|$ for any nonempty set $I$ of primes. So if $p\neq q$ are primes then $(\mathbb{Z},+,\operatorname{Val_{p}})$ and $(\mathbb{Z},+,\mathrm{Val}_{q})$ do not have a common dp-minimal expansion.

So far we have described countably many dp-minimal expansions of $(\mathbb{Z},+)$. Fact 6.5, proven by Tran and Walsberg [45], shows that there is an uncountable collection of dp-minimal expansions of $(\mathbb{Z},+)$, no two of which are interdefinable.

Fact 6.5, Fact 4.1, and dp-minimality of $(\mathbb{Z},+,<)$ together show that any expansion of $(\mathbb{Z},+)$ by a cyclic group order is dp-minimal.

It is shown in [45] that every unary definable set in every elementary extension of $(\mathbb{Z},+,C_{\alpha})$ is a finite union of tmc sets. It follows by Fact 3.8 that if $\mathscr{Z}$ expands $(\mathbb{Z},+,C_{\alpha})$ and every unary definable set in every elementary extension of $\mathscr{Z}$ is a finite union of tmc sets, then $\mathscr{Z}$ is dp-minimal. A converse is proven in [42].

In particular a dp-minimal expansion of $(\mathbb{Z},+,C_{\alpha})^{\mathrm{Sh}}$ cannot add new unary sets.

Suppose $\alpha,\beta\in\mathbb{R}/\mathbb{Z}$ are irrational and $\mathbb{Z}$-linearly independent. An easy application of Kronecker density shows that if $I$ is an infinite and co-infinite $C_{\alpha}$-convex set then $I$ is not a finite union of $C_{\beta}$-tmc sets, see [45]. Fact 6.7 follows.

We now describe a striking recent result of Alouf [1]. We first recall Fact 6.8, a special case of [19, Lemma 3.1].

Fact 6.8 shows that any dp-minimal expansion of $(\mathbb{Z},+,\operatorname{Val_{p}})$ or $(\mathbb{Z},+,C_{\alpha})$ eliminates $\exists^{\infty}$. Fact 6.9 is proven in [1].

So $(\mathbb{Z},+,<)$ is, up to interdefinibility, the only dp-minimal expansion of $(\mathbb{Z},+)$ which does not eliminate $\exists^{\infty}$. Conjecture 1 is now natural.

Johnson [21] shows that a dp-minimal expansion of a field which is not strongly minimal admits a definable non-discrete field topology. His proof makes crucial use of the fact that any dp-minimal expansion of a field eliminates $\exists^{\infty}$.

We describe new dp-minimal expansions of $(\mathbb{Z},+,C_{\alpha})$.