
Recent Posts
 Affine Images of Infinite sets
 Additive transversality of fractal sets in the reals and the integers
 Multiple ergodic averages along functions from a Hardy field: convergence, recurrence and combinatorial applications
 Distal systems and Expansive systems
 Structure of multicorrelation sequences with integer part polynomial iterates along primes
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
Posted in Analysis, Combinatorics, Ramsey Theory
Tagged Affine transformations, Bourgain, erdos, Szemeredi's theorem
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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
Posted in Combinatorics, Number Theory, paper
Tagged Furstenberg, Glasscock, Hochman, integer dimension, integer fractal, Lindenstrauss, Meiri, Peres, Richter, Shmerkin, transversality, Wu
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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
Posted in Combinatorics, Ergodic Theory, paper, Ramsey Theory
Tagged Bergelson, Hardy fields, nonconventional ergodic averages, Richter
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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 nontrivial ways. Typically, this is achieved by studying the long term behavior of the sets as the dynamics … Continue reading
Posted in Combinatorics, Probability, Ramsey Theory, Tool
Tagged Bergelson, Dodos, Kanallopoulos, Poincare, Tyros
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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
Posted in Combinatorics, paper
Tagged density, erdos, Richter, Robertson, StoneCech compactification, sumset, Ultrafilters, weak mixing
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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 IPset, i.e., there exists … Continue reading
Posted in Combinatorics, Number Theory, State of the art
Tagged Banach density, Bohr sets, density, 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 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
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
3 Comments
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
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
Posted in Analysis, Combinatorics, paper
Tagged Banach density, collinear points, Lipschitz, Pomerance, Rademacher's theorem, Richter
2 Comments