Tag Archives: Upper density

Weighted densities with multiplicative structure

The upper density of a set , defined by provides a useful way to measure subsets of . For instance, whenever , contains arbitrarily long arithmetic progressions, this is Szemerédi’s theorem. A fundamental property of the upper density is that … Continue reading

Posted in Combinatorics, Number Theory, Tool | Tagged , , , | 1 Comment

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