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Category Archives: Classic results
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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Gaussian systems
Examples of measure preserving systems with varied behaviours are vital in ergodic theory, to understand the general properties and to have counter examples to false statements. One classical method to craft examples with specific properties is the socalled Gaussian construction. … Continue reading
Posted in Classic results, Ergodic Theory
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The horocycle flow is mixing of all orders
— 1. Introduction — The main purpose of this post is to present a proof, due to Brian Marcus, that the horocycle flow is mixing of all orders. The precise definition of mixing of all orders for actions is given … Continue reading
Posted in Analysis, Classic results, Ergodic Theory
Tagged flows, horocycle flow, Marcus, mixing of all orders, van der Corput
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Primes of the form x^2+2y^2
In this post I will present a quite nice proof of the following fact from elementary number theory: Theorem 1 Let be a prime number. There are such that if and only if is a quadratic residue . Recall that … Continue reading
Posted in Classic results, Number Theory
Tagged Minkowski's theorem, polynomials, primes
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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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Disintegration of measures
In this post I will talk about conditional expectation and disintegration of a measure with respect to a algebra. All this is classical probability theory but I think not many people (me included) come across this in a standard course … Continue reading
Posted in Analysis, Classic results, Tool
Tagged conditional expectation, disintegration of measure, measure theory
6 Comments