In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a nilsequence is a type of numerical sequence playing a role in
ergodic theory
Ergodic theory (Greek: ' "work", ' "way") is a branch of mathematics that studies statistical properties of deterministic dynamical systems; it is the study of ergodicity. In this context, statistical properties means properties which are expres ...
and
additive combinatorics. The concept is related to
nilpotent Lie group
In mathematics, specifically group theory, a nilpotent group ''G'' is a group that has an upper central series that terminates with ''G''. Equivalently, its central series is of finite length or its lower central series terminates with .
Intuiti ...
s and
almost periodicity. The name arises from the part played in the theory by
compact nilmanifolds of the type
where
is a nilpotent
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
and
a
lattice in it.
The idea of a basic nilsequence defined by an element
of
and continuous function
on
is to take
, for
an integer, as
. General nilsequences are then uniform limits of basic nilsequences. For the statement of conjectures and theorems, technical side conditions and quantifications of complexity are introduced. Much of the combinatorial importance of nilsequences reflects their close connection with the
Gowers norm
In mathematics, in the field of additive combinatorics, a Gowers norm or uniformity norm is a class of norms on functions on a finite group or group-like object which quantify the amount of structure present, or conversely, the amount of randomne ...
. As explained by Host and Kra, nilsequences originate in evaluating functions on orbits in a "nilsystem"; and nilsystems are "characteristic for multiple correlations".
Case of the circle group
The
circle group
In mathematics, the circle group, denoted by \mathbb T or \mathbb S^1, is the multiplicative group of all complex numbers with absolute value 1, that is, the unit circle in the complex plane or simply the unit complex numbers.
\mathbb T = \ ...
arises as the special case of the
real line
In elementary mathematics, a number line is a picture of a graduated straight line (geometry), line that serves as visual representation of the real numbers. Every point of a number line is assumed to correspond to a real number, and every real ...
and its subgroup of the
integers. It has
nilpotency class equal to 1, being abelian, and the requirements of the general theory are to generalise to nilpotency class
The semi-open unit interval is a
fundamental domain
Given a topological space and a group acting on it, the images of a single point under the group action form an orbit of the action. A fundamental domain or fundamental region is a subset of the space which contains exactly one point from each o ...
, and for that reason the
fractional part
The fractional part or decimal part of a non‐negative real number x is the excess beyond that number's integer part. If the latter is defined as the largest integer not greater than , called floor of or \lfloor x\rfloor, its fractional part can ...
function is involved in the theory. Functions involving the fractional part
of the variable in the circle group occur, under the name "bracket polynomials". Since the theory is in the setting of
Lipschitz function
In mathematical analysis, Lipschitz continuity, named after German mathematician Rudolf Lipschitz, is a strong form of uniform continuity for functions. Intuitively, a Lipschitz continuous function is limited in how fast it can change: there exis ...
s, which are ''a fortiori'' continuous, the discontinuity of the fractional part at 0 has to be managed.
That said, the sequences
, where
is a given irrational real number, and
an integer, and studied in
diophantine approximation, are simple examples for the theory. Their construction can be thought of in terms of the skew product construction in ergodic theory, adding one dimension.
Polynomial sequences
The imaginary exponential function
maps the real numbers to the circle group (see
Euler's formula#Topological interpretation). A numerical sequence
where
is a polynomial function with real coefficients, and
is an integer variable, is a type of
trigonometric polynomial In the mathematical subfields of numerical analysis and mathematical analysis, a trigonometric polynomial is a finite linear combination of functions sin(''nx'') and cos(''nx'') with ''n'' taking on the values of one or more natural numbers. The c ...
, called a "polynomial sequence" for the purposes of the nilsequence theory. The generalisation to nilpotent groups that are not abelian relies on the
Hall–Petresco identity In mathematics, the Hall–Petresco identity (sometimes misspelled Hall–Petrescu identity) is an identity holding in any group. It was introduced by and . It can be proved using the commutator collecting process In group theory, a branch of mathe ...
from group theory for a workable theory of polynomials. In particular the polynomial sequence comes with a definite
degree
Degree may refer to:
As a unit of measurement
* Degree (angle), a unit of angle measurement
** Degree of geographical latitude
** Degree of geographical longitude
* Degree symbol (°), a notation used in science, engineering, and mathematics
...
.
Möbius function and nilsequences
A family of conjectures
was made by
Ben Green and
Terence Tao
Terence Chi-Shen Tao (; born 17 July 1975) is an Australian-American mathematician. He is a professor of mathematics at the University of California, Los Angeles (UCLA), where he holds the James and Carol Collins chair. His research includes ...
, concerning the
Möbius function of prime number theory and
-step nilsequences. Here the underlying Lie group
is assumed
simply connected
In topology, a topological space is called simply connected (or 1-connected, or 1-simply connected) if it is path-connected and every path between two points can be continuously transformed (intuitively for embedded spaces, staying within the spac ...
and nilpotent with length at most
. The nilsequences considered are of type
with some fixed
in
, and the function
continuous and taking values in . The form of the conjecture, which requires a stated metric on the nilmanifold and Lipschitz bound in the implied constant, is that the average of
up to
is smaller asymptotically than any fixed inverse power of
As a subsequent paper published in 2012 proving the conjectures put it, ''The Möbius function is strongly orthogonal to nilsequences''.
Subsequently Green, Tao and
Tamar Ziegler
Tamar Debora Ziegler (; born 1971) is an Israeli mathematician known for her work in ergodic theory, combinatorics and number theory. She holds the Henry and Manya Noskwith Chair of Mathematics at the Einstein Institute of Mathematics at the Heb ...
also proved a family
of inverse theorems for the Gowers norm, stated in terms of nilsequences. This completed a program of proving asymptotics for simultaneous prime values of linear forms.
Tao has commented in his book ''Higher Order Fourier Analysis'' on the role of nilsequences in the inverse theorem proof. The issue being to extend IG results from the
finite field
In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
case to general finite
cyclic groups, the "classical phases"—essentially the exponentials of polynomials natural for the circle group—had proved inadequate. There were options other than nilsequences, in particular direct use of bracket polynomials. But Tao writes that he prefers nilsequences for the underlying Lie theory structure.
Equivalent form for averaged Chowla and Sarnak conjectures
Tao has proved that a conjecture on nilsequences is an equivalent of an averaged form of a noted conjecture of
Sarvadaman Chowla
Sarvadaman D. S. Chowla (22 October 1907 – 10 December 1995) was an Indian American mathematician, specializing in number theory.
Early life
He was born in London, since his father, Gopal Chowla, a professor of mathematics in Lahore, was then s ...
involving only the Möbius function, and the way it self-correlates.
Peter Sarnak made a conjecture on the non-correlation of the Möbius function with more general sequences from ergodic theory, which is a consequence of Chowla's conjecture. Tao's result on averaged forms showed all three conjectures are equivalent. The 2018 paper ''The logarithmic Sarnak conjecture for ergodic weights'' by Frantzikinakis and Host used this approach to prove unconditional results on the
Liouville function.
Notes
{{reflist
Sequences and series
Nilpotent groups
Ergodic theory
Additive combinatorics