# Uniformity seminorms on ℓ∞ and applications

@article{Host2009UniformitySO, title={Uniformity seminorms on ℓ∞ and applications}, author={Bernard Host and Bryna Kra}, journal={Journal d'Analyse Math{\'e}matique}, year={2009}, volume={108}, pages={219-276} }

A key tool in recent advances in understanding arithmetic progressions and other patterns in subsets of the integers is certain norms or seminorms. One example is the norms on ℤ/Nℤ introduced by Gowers in his proof of Szemerédi’s Theorem, used to detect uniformity of subsets of the integers. Another example is the seminorms on bounded functions in a measure preserving system (associated to the averages in Furstenberg’s proof of Szemerédi’s Theorem) defined by the authors. For each integer k ≥ 1…

## 66 Citations

Quantitative inverse theory of Gowers uniformity norms

- Mathematics
- 2020

(This text is a survey written for the Bourbaki seminar on the work of F. Manners.)
Gowers uniformity norms are the central objects of higher order Fourier analysis, one of the cornerstones of…

Multiple recurrence and finding patterns in dense sets

- Mathematics
- 2014

Szemeredi’s Theorem asserts that any positive-density subset of the integers must contain arbitrarily long arithmetic progressions. It is one of the central results of additive combinatorics. After…

The logarithmic Sarnak conjecture for ergodic weights

- Mathematics
- 2017

The M\"obius disjointness conjecture of Sarnak states that the M\"obius function does not correlate with any bounded sequence of complex numbers arising from a topological dynamical system with zero…

Weighted multiple ergodic averages and correlation sequences

- MathematicsErgodic Theory and Dynamical Systems
- 2016

We study mean convergence results for weighted multiple ergodic averages defined by commuting transformations with iterates given by integer polynomials in several variables. Roughly speaking, we…

Extensions of J. Bourgain's double recurrence theorem

- Mathematics
- 2016

Ryo Moore: Extensions of J. Bourgain’s Double Recurrence Theorem (Under the direction of Idris Assani) The study of multiple recurrence averages was pioneered by Furstenberg in 1977, when he provided…

london mathematical society lecture note series

- Mathematics
- 2007

Denis Benois. Trivial zeros of p-adic L-functions and Iwasawa theory. We prove that the expected properties of Euler systems imply quite general MazurTate-Teitelbaum type formulas for derivatives of…

Higher order Fourier analysis of multiplicative functions and applications

- Mathematics
- 2014

We prove a structure theorem for multiplicative functions which states that an arbitrary bounded multiplicative function can be decomposed into two terms, one that is approximately periodic and…

A point of view on Gowers uniformity norms

- Mathematics
- 2010

Gowers norms have been studied extensively both in the direct sense, starting with a function and understanding the associated norm, and in the inverse sense, starting with the norm and deducing…

Inverting the Furstenberg correspondence

- Mathematics
- 2011

Given a sequence of subsets A_n of {0,...,n-1}, the Furstenberg correspondence principle provides a shift-invariant measure on Cantor space that encodes combinatorial information about infinitely…

Multiple correlation sequences and nilsequences

- Mathematics
- 2014

We study the structure of multiple correlation sequences defined by measure preserving actions of commuting transformations. When the iterates of the transformations are integer polynomials we prove…

## References

SHOWING 1-10 OF 20 REFERENCES

Distribution of values of bounded generalized polynomials

- Mathematics
- 2007

A generalized polynomial is a real-valued function which is obtained from conventional polynomials by the use of the operations of addition, multiplication, and taking the integer part; a generalized…

Nonconventional ergodic averages and nilmanifolds

- Mathematics
- 2005

We study the L2-convergence of two types of ergodic averages. The first is the average of a product of functions evaluated at return times along arithmetic progressions, such as the expressions…

The primes contain arbitrarily long arithmetic progressions

- Mathematics
- 2004

We prove that there are arbitrarily long arithmetic progressions of primes. There are three major ingredients. The first is Szemeredi�s theorem, which asserts that any subset of the integers of…

Multiple recurrence and nilsequences

- Mathematics
- 2005

Aiming at a simultaneous extension of Khintchine’s and Furstenberg’s Recurrence theorems, we address the question if for a measure preserving system $(X,\mathcal{X},\mu,T)$ and a set…

A new proof of Szemerédi's theorem

- Mathematics
- 2001

In 1927 van der Waerden published his celebrated theorem on arithmetic progressions, which states that if the positive integers are partitioned into finitely many classes, then at least one of these…

Piecewise-Bohr Sets of Integers and Combinatorial Number Theory

- Mathematics
- 2006

We use ergodic-theoretical tools to study various notions of “large” sets of integers which naturally arise in theory of almost periodic functions, combinatorial number theory, and dynamics. Call a…

Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold

- MathematicsErgodic Theory and Dynamical Systems
- 2004

We show that the orbit of a point on a compact nilmanifold X under the action of a polynomial sequence of translations on X is well distributed on the union of several sub-nilmanifolds of X. This…

Sur une nil-variété, les parties minimales associées à une translation sont uniquement ergodiques

- MathematicsErgodic Theory and Dynamical Systems
- 1991

Abstract We call nilmanifold every compact space X on which a connected locally compact nilpotent group acts transitively. We show that, if X is a nilmanifold and f is a continuous function on X,…

Analysis of two step nilsequences

- Mathematics
- 2007

Nilsequences arose in the study of the multiple ergodic averages associated to Furstenberg's proof of Szemer\'edi's Theorem and have since played a role in problems in additive combinatorics.…

Uniform distribution of sequences

- Mathematics
- 1974

( 1 ) {xn}z= Xn--Z_I Zin-Ztn-I is uniformly distributed mod 1, i.e., if ( 2 ) lim (1/N)A(x, N, {xn}z)-x (0x<_ 1), where A(x, N, {Xn)) denotes the number of indices n, l<=n<=N such that {x} is less…