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Tag Archives: van der Corput
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
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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On {x+y,xy} patterns in large sets of countable fields
Vitaly Bergelson and I have recently uploaded to the arXiv our joint paper `On patterns in large sets of countable fields‘. We prove a result concerning certain monochromatic structures in countable fields and a corresponding density version. Schur’s Theorem, proved … 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
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Convergence along ultrafilters – part II
This is the second of a series of two post whose aim is to prove the following recurrence theorem. Recall that a measure preserving system (shortened to m.p.s.) is a quadruple , where is a probability space and preserves the … Continue reading
Posted in Combinatorics, Ergodic Theory, Ramsey Theory
Tagged plim, ramsey theory, recurrence, Ultrafilters, van der Corput
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Convergence along ultrafilters
— 1. Introduction — On my previous post about recurrent theorems I stated Khintchine’s theorem and Sarkozy’s theorem. There I classified Khintchine’s theorem as a theorem about large intersections and Sarkozy’s theorem as a theorem about large recurrent times. This … Continue reading
Posted in Ergodic Theory, Ramsey Theory, Tool
Tagged plim, ramsey theory, recurrence, Ultrafilters, van der Corput
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