Goursat and Furstenberg-Weiss type lemmas
I’m collecting in this blog post a number of simple group-theoretic lemmas, all of the following flavour: if is a subgroup of some product of groups, then one of three things has to happen: ( too small) is contained in some proper subgroup of , or the elements of are constrained to some sort of […]
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 […]
A Host–Kra F^omega_2-system of order 5 that is not Abramov of order 5, and non-measurability of the inverse theorem for the U^6(F^n_2) norm; The structure of totally disconnected Host–Kra–Ziegler factors, and the inverse theorem for the U^k Gowers uniformity norms on finite abelian groups of bounded torsion
Asgar Jamneshan, Or Shalom, and myself have just uploaded to the arXiv our preprints “A Host–Kra -system of order 5 that is not Abramov of order 5, and non-measurability of the inverse theorem for the norm” and “The structure of totally disconnected Host–Kra–Ziegler factors, and the inverse theorem for the Gowers uniformity norms on finite […]
The inverse theorem for the U^3 Gowers uniformity norm on arbitrary finite abelian groups: Fourier-analytic and ergodic approaches
Asgar Jamneshan and myself have just uploaded to the arXiv our preprint “The inverse theorem for the Gowers uniformity norm on arbitrary finite abelian groups: Fourier-analytic and ergodic approaches“. This paper, which is a companion to another recent paper of ourselves and Or Shalom, studies the inverse theory for the third Gowers uniformity norm on […]
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