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In this paper we establish a new connection between central sets and the strong coincidence conjecture for fixed points of irreducible primitive substitutions of Pisot type. Central sets, first introduced by Furstenberg using notions from topological dynamics, constitute a special class of subsets of $ ats$ possessing strong combinatorial properties: Each central set contains arbitrarily long arithmetic progressions, and solutions to all partition regular systems of homogeneous linear equations. We give an equivalent reformulation of the strong coincidence condition in terms of central sets and minimal idempotent ultrafilters in the Stone-v{C}ech compactification $beta ats .$ This provides a new arithmetical approach to an outstanding conjecture in tiling theory, the Pisot substitution conjecture. The results in this paper rely on interactions between different areas of mathematics, some of which had not previously been directly linked: They include the general theory of combinatorics on words, abstract numeration systems, tilings, topological dynamics and the algebraic/topological properties of Stone-v{C}ech compactification of $ ats.$
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A sequential dynamical system (SDS) consists of a graph, a set of local functions and an update schedule. A linear sequential dynamical system is an SDS whose local functions are linear. In this paper, we derive an explicit closed formula for any linear SDS as a synchronous dynamical system. We also show constructively, that any synchronous linear system can be expressed as a linear SDS, i.e. it can be written as a product of linear local functions. Furthermore, we study the connection between linear SDS and the incidence algebras of partially ordered sets (posets). Specifically, we show that the M{o}bius function of any poset can be computed via an SDS, whose graph is induced by the Hasse diagram of the poset. Finally, we prove a cut theorem for the M{o}bius functions of posets with respect to certain chain decompositions.
In this paper we analyze Garden-of-Eden (GoE) states and fixed points of monotone, sequential dynamical systems (SDS). For any monotone SDS and fixed update schedule, we identify a particular set of states, each state being either a GoE state or reaching a fixed point, while both determining if a state is a GoE state and finding out all fixed points are generally hard. As a result, we show that the maximum size of their limit cycles is strictly less than ${nchoose lfloor n/2 rfloor}$. We connect these results to the Knaster-Tarski theorem and the LYM inequality. Finally, we establish that there exist monotone, parallel dynamical systems (PDS) that cannot be expressed as monotone SDS, despite the fact that the converse is always true.
Superfilters are generalized ultrafilters, which capture the underlying concept in Ramsey theoretic theorems such as van der Waerdens Theorem. We establish several properties of superfilters, which generalize both Ramseys Theorem and its variant for ultrafilters on the natural numbers. We use them to confirm a conjecture of Kov{c}inac and Di Maio, which is a generalization of a Ramsey theoretic result of Scheepers, concerning selections from open covers. Following Bergelson and Hindmans 1989 Theorem, we present a new simultaneous generalization of the theorems of Ramsey, van der Waerden, Schur, Folkman-Rado-Sanders, Rado, and others, where the colored sets can be much smaller than the full set of natural numbers.
Let ${mathfrak C}$ be a monster model of an arbitrary theory $T$, $bar alpha$ any tuple of bounded length of elements of ${mathfrak C}$, and $bar c$ an enumeration of all elements of ${mathfrak C}$. By $S_{bar alpha}({mathfrak C})$ denote the compact space of all complete types over ${mathfrak C}$ extending $tp(bar alpha/emptyset)$, and $S_{bar c}({mathfrak C})$ is defined analogously. Then $S_{bar alpha}({mathfrak C})$ and $S_{bar c}({mathfrak C})$ are naturally $Aut({mathfrak C})$-flows. We show that the Ellis groups of both these flows are of bounded size (i.e. smaller than the degree of saturation of ${mathfrak C}$), providing an explicit bound on this size. Next, we prove that these Ellis groups do not depend on the choice of the monster model ${mathfrak C}$; thus, we say that they are absolute. We also study minimal left ideals (equivalently subflows) of the Ellis semigroups of the flows $S_{bar alpha}({mathfrak C})$ and $S_{bar c}({mathfrak C})$. We give an example of a NIP theory in which the minimal left ideals are of unbounded size. We show that in each of these two cases, boundedness of a minimal left ideal is an absolute property (i.e. it does not depend on the choice of ${mathfrak C}$) and that whenever such an ideal is bounded, then its isomorphism type is also absolute. Assuming NIP, we give characterizations of when a minimal left ideal of the Ellis semigroup of $S_{bar c}({mathfrak C})$ is bounded. Then we adapt a proof of Chernikov and Simon to show that whenever such an ideal is bounded, the natural epimorphism (described by Krupinski, Pillay and Rzepecki) from the Ellis group of the flow $S_{bar c}({mathfrak C})$ to the Kim-Pillay Galois group $Gal_{KP}(T)$ is an isomorphism (in particular, $T$ is G-compact). We provide some counter-examples for $S_{bar alpha}({mathfrak C})$ in place of $S_{bar c}({mathfrak C})$.
We explore the properties of non-piecewise syndetic sets with positive upper density, which we call discordant, in countable amenable (semi)groups. Sets of this kind are involved in many questions of Ramsey theory and manifest the difference in complexity between the classical van der Waerdens theorem and Szemeredis theorem. We generalize and unify old constructions and obtain new results about these historically interesting sets. Here is a small sample of our results. $bullet$ We connect discordant sets to recurrence in dynamical systems, and in this setting we exhibit an intimate analogy between discordant sets and nowhere dense sets having positive measure. $bullet$ We introduce a wide-ranging generalization of the squarefree numbers, producing many examples of discordant sets in $mathbb{Z}$, $mathbb{Z}^d$, and the Heisenberg group. We develop a unified method to compute densities of these discordant sets. $bullet$ We show that, for any countable abelian group $G$, any F{o}lner sequence $Phi$ in $G$, and any $c in (0, 1)$, there exists a discordant set $A subseteq G$ with $d_Phi(A) = c$. Here $d_Phi$ denotes density along $Phi$. Along the way, we draw from various corners of mathematics, including classical Ramsey theory, ergodic theory, number theory, and topological and symbolic dynamics.
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