Tag Archives: Banach density

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 , , , , , | 2 Comments

Additive vs multiplicative densities

It is basic fact of measure theory that there is no uniform measure on a countable set such as the set of all natural numbers. However, there are many ways to measure size of subsets of . For instance, the … Continue reading

Posted in Combinatorics | Tagged , , , , | 1 Comment

Sets of nice recurrence

— 1. Introduction — Let be a probability space and be a (measurable) map such that the set has the same measure as the set for all (measurable) sets . We call the triple a measure preserving system. All sets … Continue reading

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Jin’s Theorem

— 1. Introduction — The PoincarĂ© recurrence theorem (or, more accurately, its proof) implies that, given a set with positive upper Banach density, i.e. then there exists some such that . In fact one gets that the set of those … Continue reading

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