# Category Archives: Analysis

## 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

## 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 | | 2 Comments

## 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 | | 22 Comments

## 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 | | 7 Comments

## 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 | | 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 | | 6 Comments

## 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