Expansions of the group of integers by Beatty sequences (Q2067513)

A Beatty sequence is a sequence \((\lfloor nr \rfloor)_{n>0}\) where \(r>1\) is an irrational real number and \(\lfloor nr \rfloor\) denotes the greatest integer which is less than or equal to \(nr\). Let \(L=\{+,-,0,c\}\) be the language of abelian groups with a distinguished element interpreted in \(\mathbb{Z}\) by \(1\) and denote by \(T\) the theory of the \(L\)-structure \((\mathbb{Z},+,-,0,1)\). The authors extend \(L\) to the language \(L_P=L\cup \{P\}\), where \(P\) is a unary predicate which is interpreted by \(P_r=\{\lfloor nr \rfloor\; :\; n\in \mathbb{Z}\backslash \{0\}\}\) in \(\mathbb{Z}\) and they let \(\mathfrak{Z}\) be the structure \((\mathbb{Z},+,-,0,1,P_r)\). They prove that every \(X\subset \mathbb{Z}\) definable in \(\mathfrak{Z}\) has the uniform gaps property, which means that there is a positive integer \(M_X\) such that \(\forall x\in X \; \left(X\cap \{x+1,\dots,x+M_X\}\neq \emptyset\right)\). It follows that the order on \(\mathbb{Z}\) and its multiplication are not definable in \(\mathfrak{Z}\). For \(\vec{k}=(k_1,\dots,k_n)\in \mathbb{Z}^n\) and \(I\subseteq \{1,\dots,n\}\) let \(\Phi_{\vec{k},I}(x_1,\dots,x_n)\) be the formula \[\exists y\left(\bigwedge_{i\in I}x_i+k_iy\in P\wedge \bigwedge_{j\notin I}x_j+k_jy\notin P\right).\] The authors define a quantifier-free formula \(\Psi_{\vec{k},I}\) such that \[\mathfrak{Z}\models (*):\; \forall x_1\dots\forall x_n\left(\Phi_{\vec{k},I}(x_1,\dots,x_n) \leftrightarrow\Psi_{\vec{k},I}(x_1,\dots,x_n)\right),\] and they denote by \(T_r\) the \(L_P\) theory which extends \(T\) with above formulas \((*)\). In a model of \(T_r\), the subgroup of elements which are divisible by \(n\) is not definable by a quantifier-free formula. We denote by \(L_{\pm}\) the language \(L_P\cup \{D_{n,+}, \; D_{n,-}\; :\; n\geq 1\}\) and by \(T_{\pm}\) the theory \(T_r\) together with the axioms \(\forall x\; \left(D_{n,+}(x)\right)\leftrightarrow \exists y\left(x=ny \wedge y\in P\right)\) and \(\forall x\; \left(D_{n,-}(x)\right)\leftrightarrow \exists y\left(x=ny \wedge y\notin P\right)\). The authors prove that \(T_{\pm}\) admits quantifier elimination. The structure \(\left(\mathbb{Z},+,-,0,1,\left(D_{n,+}^{\mathbb{Z}}\right)_{n>0},\left(D_{n,-}^{\mathbb{Z}}\right)_{n>0}\right)\) also has quantifier elimination. Extend \(T_{\pm}\) to a theory \(T_r^*\) by adding for every \(k\in\mathbb{N}\): \(kc\in P\), if \(k\in P_r\), and \(kc\notin P\), if \(k\notin P_r\). The theory \(T_r^*\) still has quantifier elimination, but furthermore \(T_r^*\) is complete. Now, \(T_r^*\) is not stable. Next, let \(\mathbb{S}\) be the group of unimodular complex numbers. It is equipped with the counter-clockwise orientation that is, \(\mathcal{O}(\alpha,\beta,\gamma)\) holds if and only if the exist real numbers \(x<y<z\) with \(z-x<1\) and \(\alpha=\exp\left(2i\pi x\right)\), \(\beta=\exp\left(2i\pi y\right)\) and \(\gamma=\exp\left(2i\pi z\right)\). We denote by \((\alpha,\beta)\) the oriented interval \(\{\gamma\in \mathbb{S}\; :\; \mathcal{O}(\alpha,\gamma,\beta)\}\). Let \(h\) be the mapping from \(\mathbb{Z}\) to \(\mathbb{S}\) defined by \(h(n)=\exp\left(2i\pi\frac{n}{r}\right)\), and by \(\Gamma_r\) its image. The image of \(P_r\) is \((h(-1),1)\cap \Gamma_r\), that we still denote by \(P_r\). Now we let \(\mathfrak{G}\) be the structure \((\Gamma_r,\cdot,^{-1},1,h(1),P_r)\). Every interval of \(\Gamma_r\) is definable in \(\mathfrak{G}\). The last result of the authors asserts that if \(X\subseteq\Gamma_r^n\) is definable in \(\mathfrak{G}\) but not in \((\Gamma_r,\cdot)\), then \((\Gamma_r,\cdot,X)\) is interdefinable with \(\mathfrak{G}\).