Category Archives: Combinatorics

Affine Images of Infinite sets

— 1. Szemerédi’s theorem as affine images — Szemerédi’s theorem is usually stated as “every set with positive upper density contains arbitrarily long arithmetic progressions”, but it can also be formulated without explicit mention of arithmetic progressions: Theorem 1 (Szemerédi’s … Continue reading

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Additive transversality of fractal sets in the reals and the integers

Daniel Glasscock, Florian Richter and I have just uploaded to the arXiv our new paper entitled “Additive transversality of fractal sets in the reals and the integers”. In this paper we introduce a new notion of fractal sets of natural … Continue reading

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Multiple ergodic averages along functions from a Hardy field: convergence, recurrence and combinatorial applications

Vitaly Bergelson, Florian Richter and I have recently uploaded to the arXiv our new paper entitled “Multiple ergodic averages along functions from a Hardy field: convergence, recurrence and combinatorial applications”. We establish a new multiple recurrence result (and consequentially a … Continue reading

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Optimal intersectivity

In ergodic Ramsey theory, one often wants to prove that certain dynamically defined sets in a probability space intersect (or “recur”) in non-trivial ways. Typically, this is achieved by studying the long term behavior of the sets as the dynamics … Continue reading

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A proof a sumset conjecture of Erdős

Florian Richter, Donald Robertson and I have uploaded to the arXiv our paper entitled A proof a sumset conjecture of Erdős. The main goal of the paper is to prove the following theorem, which verifies a conjecture of Erdős discussed … Continue reading

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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 IP-set, i.e., there exists … Continue reading

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An arithmetic van der Corput trick and the polynomial van der Waerden theorem

The van der Corput difference theorem (or van der Corput trick) was developed (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 … Continue reading

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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

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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

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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

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