
Recent Posts
 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
 SzemerĂ©di Theorem Part VI – Dichotomy between weak mixing and compact extension
 Gaussian systems
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 Banach density, collinear points, Lipschitz, Pomerance, Rademacher's theorem, Richter
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 addition and multiplication, Banach density, folner sequences, natural density, Upper density
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 Banach density, Correspondence principle, Furstenberg, nice recurrence, recurrence
Leave a comment
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