# Category Archives: Tool

## An arithmetic van der Corput trick and the polynomial van der Waerden theorem

The van der Corput difference theorem (or trick) was develop (unsurprisingly) by van der Corput, and deals with uniform distribution of sequences in the torus. Theorem 1 (van der Corput trick) Let be a sequence in a torus . If … Continue reading

## Piecewise syndetic sets, topological dynamics and ultrafilters

In this post I explore the notion of piecewise syndeticity and its relation to topological dynamical systems and the Stone-Čech compactification. I restrict attention to the additive semigroup but most results presented are true in much bigger generality (and I … Continue reading

## 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

## Factors and joinings of measure preserving systems

The joining of two measure preserving systems is a third measure preserving system that has the two original systems as factors. In analogy with classical arithmetic, using joinings it is possible to have a notion of a common multiple of … Continue reading

## Sated extensions

I recently learned of a promising technique in ergodic Ramsey theory that is useful to establish multiple recurrence and convergence of nonconventional ergodic averages. The trick is to reduce the general statement to certain systems, called sated systems. A common … Continue reading

## Banach density with respect to a single Folner sequence

In this short post I show that in any countable amenable group the (left) upper Banach density of a set can be obtained by looking only at translations of a given Følner sequence. Definitions and the precise statement are given … Continue reading

## Ergodic Decomposition

— 1. Introduction — In the study of measurable dynamics, the basic object of study is a measure preserving system: a quadruple , where is a set, is a -algebra over , is a probability measure on and is a … Continue reading