Tag Archives: Bergelson

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

Posted in Combinatorics, Ramsey Theory, Tool | Tagged , , , , , , , , , | Leave a comment

Measure preserving actions of affine semigroups and {x+y,xy} patterns

Vitaly Bergelson and I have recently submitted to the arXiv our paper entitled `Measure preserving actions of affine semigroups and patterns’. The main purpose of this paper is to extend the results of our previous paper, establishing some partial progress … Continue reading

Posted in Combinatorics, paper, Ramsey Theory | Tagged , , , , , , , | Leave a comment

New polynomial and multidimensional extensions of classical partition results

Vitaly Bergelson, John Johnson and I recently uploaded to the arXiv a paper entitled “New polynomial and multidimensional extensions of classical partition results“. In this post I will give some motivating examples for the results in the paper. To keep … Continue reading

Posted in Combinatorics, paper, Ramsey Theory | Tagged , , , , , | 1 Comment

Convergence and Recurrence of Z actions

Ergodic Ramsey Theory started with Furstenberg’s proof of Szemeredi’s theorem in arithmetic progressions in 1977. Through a correspondence principle, Furstenberg realized that Szemeredi’s theorem follows from a dynamical statement: for every invertible, ergodic measure preserving transformation of a probability space … Continue reading

Posted in Ergodic Theory, State of the art | Tagged , , , , , , , , , , , , , , | Leave a comment

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

Posted in Combinatorics, Ergodic Theory, paper, Ramsey Theory | Tagged , , , , , , , , , , | 3 Comments

Brauer’s theorem and a coloring trick of Bergelson

— 1. Introduction — Ramsey theory concerns essentially two types of results: coloring and density results. Coloring results state that given a finite partition of some structured set (usually ) one of the cells in the partition still has some … Continue reading

Posted in Combinatorics, Ergodic Theory, Ramsey Theory, Tool | Tagged , , , , , , | 3 Comments