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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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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.
In 1964, ErdH{o}s, Hajnal and Moon introduced a saturation version of Turans classical theorem in extremal graph theory. In particular, they determined the minimum number of edges in a $K_r$-free, $n$-vertex graph with the property that the addition of any further edge yields a copy of $K_r$. We consider analogues of this problem in other settings. We prove a saturation version of the ErdH{o}s-Szekeres theorem about monotone subsequences and saturati
Let $U_1, ldots, U_n$ be a collection of commuting measure preserving transformations on a probability space $(Omega, Sigma, mu)$. Associated with these measure preserving transformations is the ergodic strong maximal operator $mathsf M ^ast _{mathsf S}$ given by [ mathsf M ^ast _{mathsf S} f(omega) := sup_{0 in R subset mathbb{R}^n}frac{1}{#(R cap mathbb{Z}^n)}sum_{(j_1, ldots, j_n) in Rcap mathbb{Z}^n}big|f(U_1^{j_1}cdots U_n^{j_n}omega)big|, ] where the supremum is taken over all open rectangles in $mathbb{R}^n$ containing the origin whose sides are parallel to the coordinate axes. For $0 < alpha < 1$ we define the sharp Tauberian constant of $mathsf M ^ast _{mathsf S}$ with respect to $alpha$ by [ mathsf C ^ast _{mathsf S} (alpha) := sup_{substack{E subset Omega mu(E) > 0}}frac{1}{mu(E)}mu({omega in Omega : mathsf M ^ast _{mathsf S} chi_E (omega) > alpha}). ] Motivated by previous work of A. A. Solyanik and the authors regarding Solyanik estimates for the geometric strong maximal operator in harmonic analysis, we show that the Solyanik estimate [ lim_{alpha rightarrow 1}mathsf C ^ast _{mathsf S}(alpha) = 1 ] holds, and that in particular we have [mathsf C ^ast _{mathsf S}(alpha) - 1 lesssim_n (1 - frac{1}{alpha})^{1/n}] provided that $alpha$ is sufficiently close to $1$. Solyanik estimates for centered and uncentered ergodic Hardy-Littlewood maximal operators associated with $U_1, ldots, U_n$ are shown to hold as well. Further directions for research in the field of ergodic Solyanik estimates are also discussed.
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.$
Let $v$ be an odd real polynomial (i.e. a polynomial of the form $sum_{j=1}^ell a_jx^{2j-1}$). We utilize sets of iterated differences to establish new results about sets of the form $mathcal R(v,epsilon)={ninmathbb{N},|,|v(n)|{<epsilon}}$ where $|cdot|$ denotes the distance to the closest integer. We then apply the new diophantine results to obtain applications to ergodic theory and combinatorics. In particular, we obtain a new characterization of weakly mixing systems as well as a new variant of Furstenberg-Sarkozy theorem.
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