# Tag Archives: van der Corput

## A viewpoint on Katai’s orthogonality criterion

The Liouville function, defined as the completely multiplicative function which sends every prime to , encodes several important properties of the primes. For instance, the statement that is equivalent to the prime number theorem, while the improved (and essentially best … Continue reading

## 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

## 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

## 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

## 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