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Tag Archives: Ultrafilters
Erdős Sumset conjecture
Hindman’s finite sums theorem is one of the most famous and useful theorems in Ramsey theory. It states that for any finite partition of the natural numbers, one of the cells of this partition contains an IPset, i.e., there exists … Continue reading
Posted in Combinatorics, Number Theory, State of the art
Tagged Austin, Banach density, Bohr sets, erdos, Jin, sumset, Ultrafilters, weak mixing
2 Comments
Piecewise syndetic sets, topological dynamics and ultrafilters
In this post I explore the notion of piecewise syndeticity and its relation to topological dynamical systems and the StoneČech compactification. I restrict attention to the additive semigroup but most results presented are true in much bigger generality (and I … Continue reading
Posted in Classic results, Combinatorics, Tool, Topological Dynamics
Tagged piecewise syndetic, recurrence, Ultrafilters
2 Comments
Measure preserving actions of affine semigroups and {x+y,xy} patterns
Vitaly Bergelson and I have recently submitted to the arXiv our paper entitled `Measure preserving actions of affine semigroups and patterns’. The main purpose of this paper is to extend the results of our previous paper, establishing some partial progress … Continue reading
Double van der Waerden
— 1. Introduction — In a previous post I presented a proof of van der Waerden’s theorem on arithmetic progressions: Theorem 1 (van der Waerden, 1927) Consider a partition of the set of the natural numbers into finitely many pieces … Continue reading
Posted in Combinatorics, Ramsey Theory
Tagged arithmetic progressions, geometric progressions, Ultrafilters, van der waerden
4 Comments
Jin’s Theorem
— 1. Introduction — The Poincaré recurrence theorem (or, more accurately, its proof) implies that, given a set with positive upper Banach density, i.e. then there exists some such that . In fact one gets that the set of those … Continue reading
Weak Mixing
— 1. Introduction — When studying measure preserving systems (defined below) there are many important classes that are worth studying separately. One way to distinguish between different classes is the level of “mixing” or “randomness” of the system. In this … Continue reading
Posted in Analysis, Ergodic Theory
Tagged idempotents, minimal, Ultrafilters, van der Corput, weak mixing
2 Comments
Properties of ultrafilters and a Theorem on arithmetic combinatorics
A Theorem of Schur (one of the earliest results in Ramsey Theory) asserts that given any finite coloring of the set of natural numbers , there exist of the same color such that also has the same color. As a … Continue reading
Posted in Combinatorics, Ramsey Theory
Tagged Hindman, ramsey theory, rings, StoneCech compactification, Ultrafilters
11 Comments