
Recent Posts
 Affine Images of Infinite sets
 Additive transversality of fractal sets in the reals and the integers
 Multiple ergodic averages along functions from a Hardy field: convergence, recurrence and combinatorial applications
 Distal systems and Expansive systems
 Structure of multicorrelation sequences with integer part polynomial iterates along primes
Category Archives: Analysis
Affine Images of Infinite sets
— 1. Szemerédi’s theorem as affine images — Szemerédi’s theorem is usually stated as “every set with positive upper density contains arbitrarily long arithmetic progressions”, but it can also be formulated without explicit mention of arithmetic progressions: Theorem 1 (Szemerédi’s … Continue reading
Posted in Analysis, Combinatorics, Ramsey Theory
Tagged Affine transformations, Bourgain, erdos, Szemeredi's theorem
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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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Large subsets of discrete hypersurfaces in Z^d contain arbitrarily many collinear points
— 1. Introduction — Recently, Florian Richter and I uploaded to the arXiv our paper titled `Large subsets of discrete hypersurfaces in contain arbitrarily many collinear points’. This was the outcome of a fun project which started when we learned … Continue reading
Posted in Analysis, Combinatorics, paper
Tagged Banach density, collinear points, Lipschitz, Pomerance, Rademacher's theorem, Richter
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Convergence of continuous function
A few years back I was playing with the strong law of large numbers (essentially I wanted to understand pointwise convergence for averages over sets other than ) following Etemadi’s proof. This didn’t lead anywhere in the end (essentially because … Continue reading
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
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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
4 Comments
The Ergodic Theorem
— 1. Introduction — One can argue that (modern) ergodic theory started with the ergodic Theorem in the early 30’s. Vaguely speaking the ergodic theorem asserts that in an ergodic dynamical system (essentially a system where “everything” moves around) the … Continue reading
Posted in Analysis, Classic results, Ergodic Theory, Tool
Tagged Akcoglu, Birkhoff, del Junco, ergodic theorem, Lindenstrauss, Rohlin, Rokhlin, von Neumann
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Polya’s criterion for positive definite sequences.
1. Introduction Let be the Torus. A function can be described by it’s Fourier series. We can, more generally, consider any Borel complex measure . The Fourier coefficients then look like , where, as usual, denotes the character of associated … Continue reading
Posted in Analysis, Classic results, Tool
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