Tag Archives: Banach density

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

Posted in Combinatorics, Number Theory, State of the art | Tagged , , , , , , , | 4 Comments

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

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

Jin’s Theorem

— 1. Introduction — The Poincaré 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 , , , , , , | 4 Comments