Abstract densities and ideals of sets

Abstract upper densities are monotone and subadditive functions from the power set of positive integers to the unit real interval that generalize the upper densities used in number theory, including the upper asymptotic density, the upper Banach density, and the upper logarithmic density. We answer a question posed by G. Grekos in 2013, and prove the existence of translation invariant abstract upper densities onto the unit interval, whose null sets are precisely the family of finite sets, or the family of sequences whose series of reciprocals converge. We also show that no such density can be atomless. (More generally, these results also hold for a large class of summable ideals.

Finite Embeddability of Sets and Ultrafilters

A set A of natural numbers is finitely embeddable in another such set B if every finite subset of A has a rightward translate that is a subset of B. This notion of finite embeddability arose in combinatorial number theory, but in this paper we study it in its own right. We also study a related notion of finite embeddability of ultrafilters on the natural numbers. Among other results, we obtain connections between finite embeddability and the algebraic and topological structure of the Stone-Cech compactification of the discrete space of natural numbers. We also obtain connections with nonstandard models of arithmetic.Comment: to appear in Bulletin of the Polish Academy of Sciences, Math Serie

An elementary proof of Jin's theorem with a bound

We present a short and self-contained proof of Jin's theorem about the piecewise syndeticity of dierence sets which is entirely elementary, in the sense that no use is made of nonstandard analysis, ergodic theory, measure theory, ultralters, or other advanced tools. An explicit bound to the number of shifts that are needed to cover a thick set is provided. Precisely, we prove the following: If A and B are sets of integers having positive upper Banach densities a and b respectively, then there exists a finite set F of cardinality at most 1/ab such that (A-B) + F covers arbitrarily long intervals

Translation invariant filters and van der Waerden's Theorem

We present a self-contained proof of a strong version of van der Waerden's Theorem. By using translation invariant filters that are maximal with respect to inclusion, a simple inductive argument shows the existence of "piecewise syndetically"-many monochromatic arithmetic progressions of any length k in every finite coloring of the natural numbers. All the presented constructions are constructive in nature, in the sense that the involved maximal filters are defined by recurrence on suitable countable algebras of sets. No use of the axiom of choice or of Zorn's Lemma is needed

Iterated hyper-extensions and an idempotent ultrafilter proof of Rado’s Theorem

By using nonstandard analysis, and in particular iterated hyperextensions, we give foundations to a peculiar way of manipulating ultrafilters on the natural numbers and their pseudo-sums. The resulting formalism is suitable for applications in Ramsey theory of numbers. To illustrate the use of our technique, we give a (rather) short proof of Milliken-Taylor’s Theorem and a ultrafilter version of Rado’s Theorem about partition regularity of diophantine equations

A taste of nonstandard methods in combinatorics of numbers

By presenting the proofs of a few sample results, we introduce the reader to the use of nonstandard analysis in aspects of combinatorics of numbers

Intersections of shifted sets

We consider shifts of a set $A\subseteq\mathbb{N}$ by elements from another set $B\subseteq\mathbb{N}$, and prove intersection properties according to the relative asymptotic size of $A$ and $B$. A consequence of our main theorem is the following: If $A=\{a_n\}$ is such that $a_n=o(n^{k/k-1})$, then the $k$-recurrence set $R_k(A)=\{x\mid |A\cap(A+x)|\ge k\}$ contains the distance sets of arbitrarily large finite sets