### Abel prize awarded to Furstenberg and Margulis

Just a short post to note that this year’s Abel prize has been awarded jointly to Hillel Furstenberg and Grigory Margulis for “for pioneering the use of methods from probability and dynamics in group theory, number theory and combinatorics”. I was not involved in the decision making process of the Abel committee this year, but […]

### Furstenberg limits of the Liouville function

Given a function on the natural numbers taking values in , one can invoke the Furstenberg correspondence principle to locate a measure preserving system – a probability space together with a measure-preserving shift (or equivalently, a measure-preserving -action on ) – together with a measurable function (or “observable”) that has essentially the same statistics as […]

### An informal version of the Furstenberg correspondence principle

One of the basic objects of study in combinatorics are finite strings or infinite strings of symbols from some given alphabet , which could be either finite or infinite (but which we shall usually take to be compact). For instance, a set of natural numbers can be identified with the infinite string of s and […]

### A Fourier-free proof of the Furstenberg-Sarkozy theorem

The following result is due independently to Furstenberg and to Sarkozy: Theorem 1 (Furstenberg-Sarkozy theorem) Let , and suppose that is sufficiently large depending on . Then every subset of of density at least contains a pair for some natural numbers with . This theorem is of course similar in spirit to results such as […]

### The Furstenberg multiple recurrence theorem and finite extensions

In 1977, Furstenberg established his multiple recurrence theorem: Theorem 1 (Furstenberg multiple recurrence) Let be a measure-preserving system, thus is a probability space and is a measure-preserving bijection such that and are both measurable. Let be a measurable subset of of positive measure . Then for any , there exists such that Equivalently, there exists […]

### 254A, Lecture 15: The Furstenberg-Zimmer structure theorem and the Furstenberg recurrence theorem

In this lecture – the final one on general measure-preserving dynamics – we put together the results from the past few lectures to establish the Furstenberg-Zimmer structure theorem for measure-preserving systems, and then use this to finish the proof of the Furstenberg recurrence theorem.

### 254A, Lecture 10: The Furstenberg correspondence principle

In this lecture, we describe the simple but fundamental Furstenberg correspondence principle which connects the “soft analysis” subject of ergodic theory (in particular, recurrence theorems) with the “hard analysis” subject of combinatorial number theory (or more generally with results of “density Ramsey theory” type). Rather than try to set up the most general and abstract […]

### Abstracting induction on scales arguments

The following situation is very common in modern harmonic analysis: one has a large scale parameter (sometimes written as in the literature for some small scale parameter , or as for some large radius ), which ranges over some unbounded subset of (e.g. all sufficiently large real numbers , or all powers of two), and […]

### Value patterns of multiplicative functions and related sequences

Joni Teräväinen and I have just uploaded to the arXiv our paper “Value patterns of multiplicative functions and related sequences“, submitted to Forum of Mathematics, Sigma. This paper explores how to use recent technology on correlations of multiplicative (or nearly multiplicative functions), such as the “entropy decrement method”, in conjunction with techniques from additive combinatorics, […]

### The logarithmically averaged and non-logarithmically averaged Chowla conjectures

Let be the Liouville function, thus is defined to equal when is the product of an even number of primes, and when is the product of an odd number of primes. The Chowla conjecture asserts that has the statistics of a random sign pattern, in the sense that for all and all distinct natural numbers […]