Author Archives: Joel Moreira

About Joel Moreira

PhD Student at OSU in Mathematics. I'm portuguese.

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

Piecewise syndetic sets, topological dynamics and ultrafilters

In this post I explore the notion of piecewise syndeticity and its relation to topological dynamical systems and the Stone-Čech compactification. I restrict attention to the additive semigroup but most results presented are true in much bigger generality (and I … Continue reading

Posted in Classic results, Combinatorics, Tool, Topological Dynamics | Tagged , , | 1 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

Szemerédi Theorem Part VI – Dichotomy between weak mixing and compact extension

This is the sixth and final post in a series about Szemerédi’s theorem. In this post I complete the proof of the Multiple Recurrence Theorem, which I showed in a previous post of this series to be equivalent to Szemerédi’s … Continue reading

Posted in Ergodic Theory, Ramsey Theory | Tagged , , , | Leave a comment

Gaussian systems

Examples of measure preserving systems with varied behaviours are vital in ergodic theory, to understand the general properties and to have counter examples to false statements. One classical method to craft examples with specific properties is the so-called Gaussian construction. … Continue reading

Posted in Classic results, Ergodic Theory | Leave a comment

Alternative proofs of two classical lemmas

Two of the most fundamental tools in ergodic Ramsey theory are the mean ergodic theorem and the van der Corput trick. Both have a classical and fairly simple proof, which I have presented before in the blog. Recently I came … Continue reading

Posted in Number Theory | Leave a comment

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 , , , , | Leave a comment