
Recent Posts
 Erdős Sumset conjecture
 An arithmetic van der Corput trick and the polynomial van der Waerden theorem
 Piecewise syndetic sets, topological dynamics and ultrafilters
 Measure preserving actions of affine semigroups and {x+y,xy} patterns
 Szemerédi Theorem Part VI – Dichotomy between weak mixing and compact extension
Author Archives: Joel Moreira
Erdős Sumset conjecture
Hindman’s finite sums theorem is one of the most famous and useful theorems in Ramsey theory. It states that for any finite partition of the natural numbers, one of the cells of this partition contains an IPset, i.e., there exists … Continue reading
Posted in Combinatorics, Number Theory, State of the art
Tagged Austin, Banach density, Bohr sets, erdos, Jin, sumset, Ultrafilters, weak mixing
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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
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 StoneČech 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 piecewise syndetic, recurrence, Ultrafilters
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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
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 Compact extensions, joinings, Szemeredi's theorem, weak mixing
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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 socalled Gaussian construction. … Continue reading
Posted in Classic results, Ergodic Theory
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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
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