# Category Archives: Tool

## A viewpoint on Katai’s orthogonality criterion

The Liouville function, defined as the completely multiplicative function which sends every prime to , encodes several important properties of the primes. For instance, the statement that is equivalent to the prime number theorem, while the improved (and essentially best … Continue reading

## Entropy of measure preserving systems

A measure preserving system is a quadruple where is a set, is a -algebra, is a probability measure and is a measurable map satisfying for every . The notion of isomorphism in the category of measure preserving systems (defined, for … Continue reading

## Optimal intersectivity

In ergodic Ramsey theory, one often wants to prove that certain dynamically defined sets in a probability space intersect (or “recur”) in non-trivial ways. Typically, this is achieved by studying the long term behavior of the sets as the dynamics … Continue reading

## 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