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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 IPset, i.e., there exists … Continue reading
Posted in Combinatorics, Number Theory, State of the art
Tagged Austin, Banach density, Bohr sets, erdos, Jin, sumset, Ultrafilters, weak mixing
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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
Posted in Analysis, Combinatorics, paper
Tagged Banach density, collinear points, Lipschitz, Pomerance, Rademacher's theorem, Richter
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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
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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
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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