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Category Archives: Tool
An arithmetic van der Corput trick and the polynomial van der Waerden theorem
The van der Corput difference theorem (or trick) was develop (unsurprisingly) by van der Corput, and deals with uniform distribution of sequences in the torus. Theorem 1 (van der Corput trick) Let be a sequence in a torus . If … Continue reading
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
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Weighted densities with multiplicative structure
The upper density of a set , defined by provides a useful way to measure subsets of . For instance, whenever , contains arbitrarily long arithmetic progressions, this is Szemerédi’s theorem. A fundamental property of the upper density is that … Continue reading
Posted in Combinatorics, Number Theory, Tool
Tagged erdos, multiplicative structure, Upper density, weighted densities
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Factors and joinings of measure preserving systems
The joining of two measure preserving systems is a third measure preserving system that has the two original systems as factors. In analogy with classical arithmetic, using joinings it is possible to have a notion of a common multiple of … Continue reading
Sated extensions
I recently learned of a promising technique in ergodic Ramsey theory that is useful to establish multiple recurrence and convergence of nonconventional ergodic averages. The trick is to reduce the general statement to certain systems, called sated systems. A common … Continue reading
Banach density with respect to a single Folner sequence
In this short post I show that in any countable amenable group the (left) upper Banach density of a set can be obtained by looking only at translations of a given Følner sequence. Definitions and the precise statement are given … Continue reading
Posted in Classic results, Combinatorics, Tool
Tagged Følner sequence, upper Banach density
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Ergodic Decomposition
— 1. Introduction — In the study of measurable dynamics, the basic object of study is a measure preserving system: a quadruple , where is a set, is a algebra over , is a probability measure on and is a … Continue reading
Posted in Analysis, Classic results, Ergodic Theory, Tool
Tagged Choquet theorem, disintegration of measures, ergodic decomposition
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