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Kaisa Matomäki, Xuancheng Shao, Joni Teräväinen, and myself have just uploaded to the arXiv our preprint “Higher uniformity of arithmetic functions in short intervals I. All intervals“. This paper investigates the higher order (Gowers) uniformity of standard arithmetic functions in analytic number theory (and specifically, the Möbius function , the von Mangoldt function
, and the generalised divisor functions
) in short intervals
, where
is large and
lies in the range
for a fixed constant
(that one would like to be as small as possible). If we let
denote one of the functions
, then there is extensive literature on the estimation of short sums
Traditionally, asymptotics for such sums are expressed in terms of a “main term” of some arithmetic nature, plus an error term that is estimated in magnitude. For instance, a sum such as would be approximated in terms of a main term that vanished (or is negligible) if
is “minor arc”, but would be expressible in terms of something like a Ramanujan sum if
was “major arc”, together with an error term. We found it convenient to cancel off such main terms by subtracting an approximant
from each of the arithmetic functions
and then getting upper bounds on remainder correlations such as
- For the Möbius function
, we simply set
, as per the Möbius pseudorandomness conjecture. (One could choose a more sophisticated approximant in the presence of a Siegel zero, as I did with Joni in this recent paper, but we do not do so here.)
- For the von Mangoldt function
, we eventually went with the Cramér-Granville approximant
, where
and
.
- For the divisor functions
, we used a somewhat complicated-looking approximant
for some explicit polynomials
, chosen so that
and
have almost exactly the same sums along arithmetic progressions (see the paper for details).
The objective is then to obtain bounds on sums such as (1) that improve upon the “trivial bound” that one can get with the triangle inequality and standard number theory bounds such as the Brun-Titchmarsh inequality. For and
, the Siegel-Walfisz theorem suggests that it is reasonable to expect error terms that have “strongly logarithmic savings” in the sense that they gain a factor of
over the trivial bound for any
; for
, the Dirichlet hyperbola method suggests instead that one has “power savings” in that one should gain a factor of
over the trivial bound for some
. In the case of the Möbius function
, there is an additional trick (introduced by Matomäki and Teräväinen) that allows one to lower the exponent
somewhat at the cost of only obtaining “weakly logarithmic savings” of shape
for some small
.
Our main estimates on sums of the form (1) work in the following ranges:
- For
, one can obtain strongly logarithmic savings on (1) for
, and power savings for
.
- For
, one can obtain weakly logarithmic savings for
.
- For
, one can obtain power savings for
.
- For
, one can obtain power savings for
.
Conjecturally, one should be able to obtain power savings in all cases, and lower down to zero, but the ranges of exponents and savings given here seem to be the limit of current methods unless one assumes additional hypotheses, such as GRH. The
result for correlation against Fourier phases
was established previously by Zhan, and the
result for such phases and
was established previously by by Matomäki and Teräväinen.
By combining these results with tools from additive combinatorics, one can obtain a number of applications:
- Direct insertion of our bounds in the recent work of Kanigowski, Lemanczyk, and Radziwill on the prime number theorem on dynamical systems that are analytic skew products gives some improvements in the exponents there.
- We can obtain a “short interval” version of a multiple ergodic theorem along primes established by Frantzikinakis-Host-Kra and Wooley-Ziegler, in which we average over intervals of the form
rather than
.
- We can obtain a “short interval” version of the “linear equations in primes” asymptotics obtained by Ben Green, Tamar Ziegler, and myself in this sequence of papers, where the variables in these equations lie in short intervals
rather than long intervals such as
.
We now briefly discuss some of the ingredients of proof of our main results. The first step is standard, using combinatorial decompositions (based on the Heath-Brown identity and (for the result) the Ramaré identity) to decompose
into more tractable sums of the following types:
- Type
sums, which are basically of the form
for some weights
of controlled size and some cutoff
that is not too large;
- Type
sums, which are basically of the form
for some weights
,
of controlled size and some cutoffs
that are not too close to
or to
;
- Type
sums, which are basically of the form
for some weights
of controlled size and some cutoff
that is not too large.
The precise ranges of the cutoffs depend on the choice of
; our methods fail once these cutoffs pass a certain threshold, and this is the reason for the exponents
being what they are in our main results.
The Type sums involving nilsequences can be treated by methods similar to those in this previous paper of Ben Green and myself; the main innovations are in the treatment of the Type
and Type
sums.
For the Type sums, one can split into the “abelian” case in which (after some Fourier decomposition) the nilsequence
is basically of the form
, and the “non-abelian” case in which
is non-abelian and
exhibits non-trivial oscillation in a central direction. In the abelian case we can adapt arguments of Matomaki and Shao, which uses Cauchy-Schwarz and the equidistribution properties of polynomials to obtain good bounds unless
is “major arc” in the sense that it resembles (or “pretends to be”)
for some Dirichlet character
and some frequency
, but in this case one can use classical multiplicative methods to control the correlation. It turns out that the non-abelian case can be treated similarly. After applying Cauchy-Schwarz, one ends up analyzing the equidistribution of the four-variable polynomial sequence
For the type sum, a model sum to study is
In a sequel to this paper (currently in preparation), we will obtain analogous results for almost all intervals with
in the range
, in which we will be able to lower
all the way to
.
Kaisa Matomäki, Maksym Radziwill, Joni Teräväinen, Tamar Ziegler and I have uploaded to the arXiv our paper Higher uniformity of bounded multiplicative functions in short intervals on average. This paper (which originated from a working group at an AIM workshop on Sarnak’s conjecture) focuses on the local Fourier uniformity conjecture for bounded multiplicative functions such as the Liouville function . One form of this conjecture is the assertion that
The conjecture gets more difficult as increases, and also becomes more difficult the more slowly
grows with
. The
conjecture is equivalent to the assertion
For , the conjecture is equivalent to the assertion
Now we apply the same strategy to (4). For abelian the claim follows easily from (3), so we focus on the non-abelian case. One now has a polynomial sequence
attached to many
, and after a somewhat complicated adaptation of the above arguments one again ends up with an approximate functional equation
We give two applications of this higher order Fourier uniformity. One regards the growth of the number
The second application is to obtain cancellation for various polynomial averages involving the Liouville function or von Mangoldt function
, such as
A sequence of complex numbers is said to be quasiperiodic if it is of the form
for some real numbers and continuous function
. For instance, linear phases such as
(where
) are examples of quasiperiodic sequences; the top order coefficient
(modulo
) can be viewed as a “frequency” of the integers, and an element of the Pontryagin dual
of the integers. Any periodic sequence is also quasiperiodic (taking
and
to be the reciprocal of the period). A sequence is said to be almost periodic if it is the uniform limit of quasiperiodic sequences. For instance any Fourier series of the form
with real numbers and
an absolutely summable sequence of complex coefficients, will be almost periodic.
These sequences arise in various “complexity one” problems in arithmetic combinatorics and ergodic theory. For instance, if is a measure-preserving system – a probability space
equipped with a measure-preserving shift, and
are bounded measurable functions, then the correlation sequence
can be shown to be an almost periodic sequence, plus an error term which is “null” in the sense that it has vanishing uniform density:
This can be established in a number of ways, for instance by writing as the Fourier coefficients of the spectral measure of the shift
with respect to the functions
, and then decomposing that measure into pure point and continuous components.
In the last two decades or so, it has become clear that there are natural higher order versions of these concepts, in which linear polynomials such as are replaced with higher degree counterparts. The most obvious candidates for these counterparts would be the polynomials
, but this turns out to not be a complete set of higher degree objects needed for the theory. Instead, the higher order versions of quasiperiodic and almost periodic sequences are now known as basic nilsequences and nilsequences respectively, while the higher order version of a linear phase is a nilcharacter; each nilcharacter then has a symbol that is a higher order generalisation of the concept of a frequency (and the collection of all symbols forms a group that can be viewed as a higher order version of the Pontryagin dual of
). The theory of these objects is spread out in the literature across a number of papers; in particular, the theory of nilcharacters is mostly developed in Appendix E of this 116-page paper of Ben Green, Tamar Ziegler, and myself, and is furthermore written using nonstandard analysis and treating the more general setting of higher dimensional sequences. I therefore decided to rewrite some of that material in this blog post, in the simpler context of the qualitative asymptotic theory of one-dimensional nilsequences and nilcharacters rather than the quantitative single-scale theory that is needed for combinatorial applications (and which necessitated the use of nonstandard analysis in the previous paper).
For technical reasons (having to do with the non-trivial topological structure on nilmanifolds), it will be convenient to work with vector-valued sequences, that take values in a finite-dimensional complex vector space rather than in
. By doing so, the space of sequences is now, technically, no longer a ring, as the operations of addition and multiplication on vector-valued sequences become ill-defined. However, we can still take complex conjugates of a sequence, and add sequences taking values in the same vector space
, and for sequences taking values in different vector spaces
,
, we may utilise the tensor product
, which we will normalise by defining
This product is associative and bilinear, and also commutative up to permutation of the indices. It also interacts well with the Hermitian norm
since we have .
The traditional definition of a basic nilsequence (as defined for instance by Bergelson, Host, and Kra) is as follows:
Definition 1 (Basic nilsequence, first definition) A nilmanifold of step at most
is a quotient
, where
is a connected, simply connected nilpotent Lie group of step at most
(thus, all
-fold commutators vanish) and
is a discrete cocompact lattice in
. A basic nilsequence of degree at most
is a sequence of the form
, where
,
, and
is a continuous function.
For instance, it is not difficult using this definition to show that a sequence is a basic nilsequence of degree at most if and only if it is quasiperiodic. The requirement that
be simply connected can be easily removed if desired by passing to a universal cover, but it is technically convenient to assume it (among other things, it allows for a well-defined logarithm map that obeys the Baker-Campbell-Hausdorff formula). When one wishes to perform a more quantitative analysis of nilsequences (particularly when working on a “single scale”. sich as on a single long interval
), it is common to impose additional regularity conditions on the function
, such as Lipschitz continuity or smoothness, but ordinary continuity will suffice for the qualitative discussion in this blog post.
Nowadays, and particularly when one needs to understand the “single-scale” equidistribution properties of nilsequences, it is more convenient (as is for instance done in this ICM paper of Green) to use an alternate definition of a nilsequence as follows.
Definition 2 Let
. A filtered group of degree at most
is a group
together with a sequence
of subgroups
with
and
for
. A polynomial sequence
into a filtered group is a function such that
for all
and
, where
is the difference operator. A filtered nilmanifold of degree at most
is a quotient
, where
is a filtered group of degree at most
such that
and all of the subgroups
are connected, simply connected nilpotent filtered Lie group, and
is a discrete cocompact subgroup of
such that
is a discrete cocompact subgroup of
. A basic nilsequence of degree at most
is a sequence of the form
, where
is a polynomial sequence,
is a filtered nilmanifold of degree at most
, and
is a continuous function which is
-automorphic, in the sense that
for all
and
.
One can easily identify a -automorphic function on
with a function on
, but there are some (very minor) advantages to working on the group
instead of the quotient
, as it becomes slightly easier to modify the automorphy group
when needed. (But because the action of
on
is free, one can pass from
-automorphic functions on
to functions on
with very little difficulty.) The main reason to work with polynomial sequences
rather than geometric progressions
is that they form a group, a fact essentially established by by Lazard and Leibman; see Corollary B.4 of this paper of Green, Ziegler, and myself for a proof in the filtered group setting.
It is easy to see that any sequence that is a basic nilsequence of degree at most in the sense of the first definition, is also a basic nilsequence of degree at most
in the second definition, since a nilmanifold of degree at most
can be filtered using the lower central series, and any linear sequence
will be a polynomial sequence with respect to that filtration. The converse implication is a little trickier, but still not too hard to show: see Appendix C of this paper of Ben Green, Tamar Ziegler, and myself. There are two key examples of basic nilsequences to keep in mind. The first are the polynomially quasiperiodic sequences
where are polynomials of degree at most
, and
is a
-automorphic (i.e.,
-periodic) continuous function. The map
defined by
is a polynomial map of degree at most
, if one filters
by defining
to equal
when
, and
for
. The torus
then becomes a filtered nilmanifold of degree at most
, and
is thus a basic nilsequence of degree at most
as per the second definition. It is also possible explicitly describe
as a basic nilsequence of degree at most
as per the first definition, for instance (in the
case) by taking
to be the space of upper triangular unipotent
real matrices, and
the subgroup with integer coefficients; we leave the details to the interested reader.
The other key example is a sequence of the form
where are real numbers,
denotes the fractional part of
, and and
is a
-automorphic continuous function that vanishes in a neighbourhood of
. To describe this as a nilsequence, we use the nilpotent connected, simply connected degree
, Heisenberg group
with the lower central series filtration ,
, and
for
,
to be the discrete compact subgroup
to be the polynomial sequence
and to be the
-automorphic function
one easily verifies that this function is indeed -automorphic, and it is continuous thanks to the vanishing properties of
. Also we have
, so
is a basic nilsequence of degree at most
. One can concoct similar examples with
replaced by other “bracket polynomials” of
; for instance
will be a basic nilsequence if now vanishes in a neighbourhood of
rather than
. See this paper of Bergelson and Leibman for more discussion of bracket polynomials (also known as generalised polynomials) and their relationship to nilsequences.
A nilsequence of degree at most is defined to be a sequence that is the uniform limit of basic nilsequences of degree at most
. Thus for instance a sequence is a nilsequence of degree at most
if and only if it is almost periodic, while a sequence is a nilsequence of degree at most
if and only if it is constant. Such objects arise in higher order recurrence: for instance, if
are integers,
is a measure-preserving system, and
, then it was shown by Leibman that the sequence
is equal to a nilsequence of degree at most , plus a null sequence. (The special case when the measure-preserving system was ergodic and
for
was previously established by Bergelson, Host, and Kra.) Nilsequences also arise in the inverse theory of the Gowers uniformity norms, as discussed for instance in this previous post.
It is easy to see that a sequence is a basic nilsequence of degree at most
if and only if each of its
components are. The scalar basic nilsequences
of degree
are easily seen to form a
-algebra (that is to say, they are a complex vector space closed under pointwise multiplication and complex conjugation), which implies similarly that vector-valued basic nilsequences
of degree at most
form a complex vector space closed under complex conjugation for each
, and that the tensor product of any two basic nilsequences of degree at most
is another basic nilsequence of degree at most
. Similarly with “basic nilsequence” replaced by “nilsequence” throughout.
Now we turn to the notion of a nilcharacter, as defined in this paper of Ben Green, Tamar Ziegler, and myself:
Definition 3 (Nilcharacters) Let
. A sub-nilcharacter of degree
is a basic nilsequence
of degree at most
, such that
obeys the additional modulation property
for all
and
, where
is a continuous homomorphism
. (Note from (1) and
-automorphy that unless
vanishes identically,
must map
to
, thus without loss of generality one can view
as an element of the Pontryagial dual of the torus
.) If in addition one has
for all
, we call
a nilcharacter of degree
.
In the degree one case , the only sub-nilcharacters are of the form
for some vector
and
, and this is a nilcharacter if
is a unit vector. Similarly, in higher degree, any sequence of the form
, where
is a vector and
is a polynomial of degree at most
, is a sub-nilcharacter of degree
, and a character if
is a unit vector. A nilsequence of degree at most
is automatically a sub-nilcharacter of degree
, and a nilcharacter if it is of magnitude
. A further example of a nilcharacter is provided by the two-dimensional sequence
defined by
where are continuous,
-automorphic functions that vanish on a neighbourhood of
and
respectively, and which form a partition of unity in the sense that
for all . Note that one needs both
and
to be not identically zero in order for all these conditions to be satisfied; it turns out (for topological reasons) that there is no scalar nilcharacter that is “equivalent” to this nilcharacter in a sense to be defined shortly. In some literature, one works exclusively with sub-nilcharacters rather than nilcharacters, however the former space contains zero-divisors, which is a little annoying technically. Nevertheless, both nilcharacters and sub-nilcharacters generate the same set of “symbols” as we shall see later.
We claim that every degree sub-nilcharacter
can be expressed in the form
, where
is a degree
nilcharacter, and
is a linear transformation. Indeed, by scaling we may assume
where
uniformly. Using partitions of unity, one can find further functions
also obeying (1) for the same character
such that
is non-zero; by dividing out the
by the square root of this quantity, and then multiplying by
, we may assume that
and then
becomes a degree nilcharacter that contains
amongst its components, giving the claim.
As we shall show below, nilsequences can be approximated uniformly by linear combinations of nilcharacters, in much the same way that quasiperiodic or almost periodic sequences can be approximated uniformly by linear combinations of linear phases. In particular, nilcharacters can be used as “obstructions to uniformity” in the sense of the inverse theory of the Gowers uniformity norms.
The space of degree nilcharacters forms a semigroup under tensor product, with the constant sequence
as the identity. One can upgrade this semigroup to an abelian group by quotienting nilcharacters out by equivalence:
Definition 4 Let
. We say that two degree
nilcharacters
,
are equivalent if
is equal (as a sequence) to a basic nilsequence of degree at most
. (We will later show that this is indeed an equivalence relation.) The equivalence class
of such a nilcharacter will be called the symbol of that nilcharacter (in analogy to the symbol of a differential or pseudodifferential operator), and the collection of such symbols will be denoted
.
As we shall see below the fold, has the structure of an abelian group, and enjoys some nice “symbol calculus” properties; also, one can view symbols as precisely describing the obstruction to equidistribution for nilsequences. For
, the group is isomorphic to the Ponytragin dual
of the integers, and
for
should be viewed as higher order generalisations of this Pontryagin dual. In principle, this group can be explicitly described for all
, but the theory rapidly gets complicated as
increases (much as the classification of nilpotent Lie groups or Lie algebras of step
rapidly gets complicated even for medium-sized
such as
or
). We will give an explicit description of the
case here. There is however one nice (and non-trivial) feature of
for
– it is not just an abelian group, but is in fact a vector space over the rationals
!
Note: this post is of a particularly technical nature, in particular presuming familiarity with nilsequences, nilsystems, characteristic factors, etc., and is primarily intended for experts.
As mentioned in the previous post, Ben Green, Tamar Ziegler, and myself proved the following inverse theorem for the Gowers norms:
Theorem 1 (Inverse theorem for Gowers norms) Let
and
be integers, and let
. Suppose that
is a function supported on
such that
Then there exists a filtered nilmanifold
of degree
and complexity
, a polynomial sequence
, and a Lipschitz function
of Lipschitz constant
such that
This result was conjectured earlier by Ben Green and myself; this conjecture was strongly motivated by an analogous inverse theorem in ergodic theory by Host and Kra, which we formulate here in a form designed to resemble Theorem 1 as closely as possible:
Theorem 2 (Inverse theorem for Gowers-Host-Kra seminorms) Let
be an integer, and let
be an ergodic, countably generated measure-preserving system. Suppose that one has
for all non-zero
(all
spaces are real-valued in this post). Then
is an inverse limit (in the category of measure-preserving systems, up to almost everywhere equivalence) of ergodic degree
nilsystems, that is to say systems of the form
for some degree
filtered nilmanifold
and a group element
that acts ergodically on
.
It is a natural question to ask if there is any logical relationship between the two theorems. In the finite field category, one can deduce the combinatorial inverse theorem from the ergodic inverse theorem by a variant of the Furstenberg correspondence principle, as worked out by Tamar Ziegler and myself, however in the current context of -actions, the connection is less clear.
One can split Theorem 2 into two components:
Theorem 3 (Weak inverse theorem for Gowers-Host-Kra seminorms) Let
be an integer, and let
be an ergodic, countably generated measure-preserving system. Suppose that one has
for all non-zero
, where
. Then
is a factor of an inverse limit of ergodic degree
nilsystems.
Theorem 4 (Pro-nilsystems closed under factors) Let
be an integer. Then any factor of an inverse limit of ergodic degree
nilsystems, is again an inverse limit of ergodic degree
nilsystems.
Indeed, it is clear that Theorem 2 implies both Theorem 3 and Theorem 4, and conversely that the two latter theorems jointly imply the former. Theorem 4 is, in principle, purely a fact about nilsystems, and should have an independent proof, but this is not known; the only known proofs go through the full machinery needed to prove Theorem 2 (or the closely related theorem of Ziegler). (However, the fact that a factor of a nilsystem is again a nilsystem was established previously by Parry.)
The purpose of this post is to record a partial implication in reverse direction to the correspondence principle:
As mentioned at the start of the post, a fair amount of familiarity with the area is presumed here, and some routine steps will be presented with only a fairly brief explanation.
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
s formed by the indicator of
, e.g. the even numbers can be identified with the string
from the alphabet
, the multiples of three can be identified with the string
, and so forth. One can also consider doubly infinite strings
, which among other things can be used to describe arbitrary subsets of integers.
On the other hand, the basic object of study in dynamics (and in related fields, such as ergodic theory) is that of a dynamical system , that is to say a space
together with a shift map
(which is often assumed to be invertible, although one can certainly study non-invertible dynamical systems as well). One often adds additional structure to this dynamical system, such as topological structure (giving rise topological dynamics), measure-theoretic structure (giving rise to ergodic theory), complex structure (giving rise to complex dynamics), and so forth. A dynamical system gives rise to an action of the natural numbers
on the space
by using the iterates
of
for
; if
is invertible, we can extend this action to an action of the integers
on the same space. One can certainly also consider dynamical systems whose underlying group (or semi-group) is something other than
or
(e.g. one can consider continuous dynamical systems in which the evolution group is
), but we will restrict attention to the classical situation of
or
actions here.
There is a fundamental correspondence principle connecting the study of strings (or subsets of natural numbers or integers) with the study of dynamical systems. In one direction, given a dynamical system , an observable
taking values in some alphabet
, and some initial datum
, we can first form the forward orbit
of
, and then observe this orbit using
to obtain an infinite string
. If the shift
in this system is invertible, one can extend this infinite string into a doubly infinite string
. Thus we see that every quadruplet
consisting of a dynamical system
, an observable
, and an initial datum
creates an infinite string.
Example 1 If
is the three-element set
with the shift map
,
is the observable that takes the value
at the residue class
and zero at the other two classes, and one starts with the initial datum
, then the observed string
becomes the indicator
of the multiples of three.
In the converse direction, every infinite string in some alphabet
arises (in a decidedly non-unique fashion) from a quadruple
in the above fashion. This can be easily seen by the following “universal” construction: take
to be the set
of infinite strings
in the alphabet
, let
be the shift map
let be the observable
and let be the initial point
Then one easily sees that the observed string is nothing more than the original string
. Note also that this construction can easily be adapted to doubly infinite strings by using
instead of
, at which point the shift map
now becomes invertible. An important variant of this construction also attaches an invariant probability measure to
that is associated to the limiting density of various sets associated to the string
, and leads to the Furstenberg correspondence principle, discussed for instance in these previous blog posts. Such principles allow one to rigorously pass back and forth between the combinatorics of strings and the dynamics of systems; for instance, Furstenberg famously used his correspondence principle to demonstrate the equivalence of Szemerédi’s theorem on arithmetic progressions with what is now known as the Furstenberg multiple recurrence theorem in ergodic theory.
In the case when the alphabet is the binary alphabet
, and (for technical reasons related to the infamous non-injectivity
of the decimal representation system) the string
does not end with an infinite string of
s, then one can reformulate the above universal construction by taking
to be the interval
,
to be the doubling map
,
to be the observable that takes the value
on
and
on
(that is,
is the first binary digit of
), and
is the real number
(that is,
in binary).
The above universal construction is very easy to describe, and is well suited for “generic” strings that have no further obvious structure to them, but it often leads to dynamical systems that are much larger and more complicated than is actually needed to produce the desired string
, and also often obscures some of the key dynamical features associated to that sequence. For instance, to generate the indicator
of the multiples of three that were mentioned previously, the above universal construction requires an uncountable space
and a dynamics which does not obviously reflect the key features of the sequence such as its periodicity. (Using the unit interval model, the dynamics arise from the orbit of
under the doubling map, which is a rather artificial way to describe the indicator function of the multiples of three.)
A related aesthetic objection to the universal construction is that of the four components of the quadruplet
used to generate the sequence
, three of the components
are completely universal (in that they do not depend at all on the sequence
), leaving only the initial datum
to carry all the distinctive features of the original sequence. While there is nothing wrong with this mathematically, from a conceptual point of view it would make sense to make all four components of the quadruplet to be adapted to the sequence, in order to take advantage of the accumulated intuition about various special dynamical systems (and special observables), not just special initial data.
One step in this direction can be made by restricting to the orbit
of the initial datum
(actually for technical reasons it is better to restrict to the topological closure
of this orbit, in order to keep
compact). For instance, starting with the sequence
, the orbit now consists of just three points
,
,
, bringing the system more in line with the example in Example 1. Technically, this is the “optimal” representation of the sequence by a quadruplet
, because any other such representation
is a factor of this representation (in the sense that there is a unique map
with
,
, and
). However, from a conceptual point of view this representation is still somewhat unsatisfactory, given that the elements of the system
are interpreted as infinite strings rather than elements of a more geometrically or algebraically rich object (e.g. points in a circle, torus, or other homogeneous space).
For general sequences , locating relevant geometric or algebraic structure in a dynamical system generating that sequence is an important but very difficult task (see e.g. this paper of Host and Kra, which is more or less devoted to precisely this task in the context of working out what component of a dynamical system controls the multiple recurrence behaviour of that system). However, for specific examples of sequences
, one can use an informal procedure of educated guesswork in order to produce a more natural-looking quadruple
that generates that sequence. This is not a particularly difficult or deep operation, but I found it very helpful in internalising the intuition behind the correspondence principle. Being non-rigorous, this procedure does not seem to be emphasised in most presentations of the correspondence principle, so I thought I would describe it here.
In Notes 5, we saw that the Gowers uniformity norms on vector spaces in high characteristic were controlled by classical polynomial phases
.
Now we study the analogous situation on cyclic groups . Here, there is an unexpected surprise: the polynomial phases (classical or otherwise) are no longer sufficient to control the Gowers norms
once
exceeds
. To resolve this problem, one must enlarge the space of polynomials to a larger class. It turns out that there are at least three closely related options for this class: the local polynomials, the bracket polynomials, and the nilsequences. Each of the three classes has its own strengths and weaknesses, but in my opinion the nilsequences seem to be the most natural class, due to the rich algebraic and dynamical structure coming from the nilpotent Lie group undergirding such sequences. For reasons of space we shall focus primarily on the nilsequence viewpoint here.
Traditionally, nilsequences have been defined in terms of linear orbits on nilmanifolds
; however, in recent years it has been realised that it is convenient for technical reasons (particularly for the quantitative “single-scale” theory) to generalise this setup to that of polynomial orbits
, and this is the perspective we will take here.
A polynomial phase on a finite abelian group
is formed by starting with a polynomial
to the unit circle, and then composing it with the exponential function
. To create a nilsequence
, we generalise this construction by starting with a polynomial
into a nilmanifold
, and then composing this with a Lipschitz function
. (The Lipschitz regularity class is convenient for minor technical reasons, but one could also use other regularity classes here if desired.) These classes of sequences certainly include the polynomial phases, but are somewhat more general; for instance, they almost include bracket polynomial phases such as
. (The “almost” here is because the relevant functions
involved are only piecewise Lipschitz rather than Lipschitz, but this is primarily a technical issue and one should view bracket polynomial phases as “morally” being nilsequences.)
In these notes we set out the basic theory for these nilsequences, including their equidistribution theory (which generalises the equidistribution theory of polynomial flows on tori from Notes 1) and show that they are indeed obstructions to the Gowers norm being small. This leads to the inverse conjecture for the Gowers norms that shows that the Gowers norms on cyclic groups are indeed controlled by these sequences.
Ben Green, and I have just uploaded to the arXiv our paper “An arithmetic regularity lemma, an associated counting lemma, and applications“, submitted (a little behind schedule) to the 70th birthday conference proceedings for Endre Szemerédi. In this paper we describe the general-degree version of the arithmetic regularity lemma, which can be viewed as the counterpart of the Szemerédi regularity lemma, in which the object being regularised is a function on a discrete interval
rather than a graph, and the type of patterns one wishes to count are additive patterns (such as arithmetic progressions
) rather than subgraphs. Very roughly speaking, this regularity lemma asserts that all such functions can be decomposed as a degree
nilsequence (or more precisely, a variant of a nilsequence that we call an virtual irrational nilsequence), plus a small error, plus a third error which is extremely tiny in the Gowers uniformity norm
. In principle, at least, the latter two errors can be readily discarded in applications, so that the regularity lemma reduces many questions in additive combinatorics to questions concerning (virtual irrational) nilsequences. To work with these nilsequences, we also establish a arithmetic counting lemma that gives an integral formula for counting additive patterns weighted by such nilsequences.
The regularity lemma is a manifestation of the “dichotomy between structure and randomness”, as discussed for instance in my ICM article or FOCS article. In the degree case
, this result is essentially due to Green. It is powered by the inverse conjecture for the Gowers norms, which we and Tamar Ziegler have recently established (paper to be forthcoming shortly; the
case of our argument is discussed here). The counting lemma is established through the quantitative equidistribution theory of nilmanifolds, which Ben and I set out in this paper.
The regularity and counting lemmas are designed to be used together, and in the paper we give three applications of this combination. Firstly, we give a new proof of Szemerédi’s theorem, which proceeds via an energy increment argument rather than a density increment one. Secondly, we establish a conjecture of Bergelson, Host, and Kra, namely that if has density
, and
, then there exist
shifts
for which
contains at least
arithmetic progressions of length
of spacing
. (The
case of this conjecture was established earlier by Green; the
case is false, as was shown by Ruzsa in an appendix to the Bergelson-Host-Kra paper.) Thirdly, we establish a variant of a recent result of Gowers-Wolf, showing that the true complexity of a system of linear forms over
indeed matches the conjectured value predicted in their first paper.
In all three applications, the scheme of proof can be described as follows:
- Apply the arithmetic regularity lemma, and decompose a relevant function
into three pieces,
.
- The uniform part
is so tiny in the Gowers uniformity norm that its contribution can be easily dealt with by an appropriate “generalised von Neumann theorem”.
- The contribution of the (virtual, irrational) nilsequence
can be controlled using the arithmetic counting lemma.
- Finally, one needs to check that the contribution of the small error
does not overwhelm the main term
. This is the trickiest bit; one often needs to use the counting lemma again to show that one can find a set of arithmetic patterns for
that is so sufficiently “equidistributed” that it is not impacted by the small error.
To illustrate the last point, let us give the following example. Suppose we have a set of some positive density (say
) and we have managed to prove that
contains a reasonable number of arithmetic progressions of length
(say), e.g. it contains at least
such progressions. Now we perturb
by deleting a small number, say
, elements from
to create a new set
. Can we still conclude that the new set
contains any arithmetic progressions of length
?
Unfortunately, the answer could be no; conceivably, all of the arithmetic progressions in
could be wiped out by the
elements removed from
, since each such element of
could be associated with up to
(or even
) arithmetic progressions in
.
But suppose we knew that the arithmetic progressions in
were equidistributed, in the sense that each element in
belonged to the same number of such arithmetic progressions, namely
. Then each element deleted from
only removes at most
progressions, and so one can safely remove
elements from
and still retain some arithmetic progressions. The same argument works if the arithmetic progressions are only approximately equidistributed, in the sense that the number of progressions that a given element
belongs to concentrates sharply around its mean (for instance, by having a small variance), provided that the equidistribution is sufficiently strong. Fortunately, the arithmetic regularity and counting lemmas are designed to give precisely such a strong equidistribution result.
A succinct (but slightly inaccurate) summation of the regularity+counting lemma strategy would be that in order to solve a problem in additive combinatorics, it “suffices to check it for nilsequences”. But this should come with a caveat, due to the issue of the small error above; in addition to checking it for nilsequences, the answer in the nilsequence case must be sufficiently “dispersed” in a suitable sense, so that it can survive the addition of a small (but not completely negligible) perturbation.
One last “production note”. Like our previous paper with Emmanuel Breuillard, we used Subversion to write this paper, which turned out to be a significant efficiency boost as we could work on different parts of the paper simultaneously (this was particularly important this time round as the paper was somewhat lengthy and complicated, and there was a submission deadline). When doing so, we found it convenient to split the paper into a dozen or so pieces (one for each section of the paper, basically) in order to avoid conflicts, and to help coordinate the writing process. I’m also looking into git (a more advanced version control system), and am planning to use it for another of my joint projects; I hope to be able to comment on the relative strengths of these systems (and with plain old email) in the future.
Ben Green and I have just uploaded to the arXiv our paper, “The Möbius function is asymptotically orthogonal to nilsequences“, which is a sequel to our earlier paper “The quantitative behaviour of polynomial orbits on nilmanifolds“, which I talked about in this post. In this paper, we apply our previous results on quantitative equidistribution polynomial orbits in nilmanifolds to settle the Möbius and nilsequences conjecture from our earlier paper, as part of our program to detect and count solutions to linear equations in primes. (The other major plank of that program, namely the inverse conjecture for the Gowers norm, remains partially unresolved at present.) Roughly speaking, this conjecture asserts the asymptotic orthogonality
(1)
between the Möbius function and any Lipschitz nilsequence f(n), by which we mean a sequence of the form
for some orbit
in a nilmanifold
, and some Lipschitz function
on that nilmanifold. (The implied constant can depend on the nilmanifold and on the Lipschitz constant of F, but it is important that it be independent of the generator g of the orbit or the base point x.) The case when f is constant is essentially the prime number theorem; the case when f is periodic is essentially the prime number theorem in arithmetic progressions. The case when f is almost periodic (e.g.
for some irrational
) was established by Davenport, using the method of Vinogradov. The case when f was a 2-step nilsequence (such as the quadratic phase
; bracket quadratic phases such as
can also be covered by an approximation argument, though the logarithmic decay in (1) is weakened as a consequence) was done by Ben and myself a few years ago, by a rather ad hoc adaptation of Vinogradov’s method. By using the equidistribution theory of nilmanifolds, we were able to apply Vinogradov’s method more systematically, and in fact the proof is relatively short (20 pages), although it relies on the 64-page predecessor paper on equidistribution. I’ll talk a little bit more about the proof after the fold.
There is an amusing way to interpret the conjecture (using the close relationship between nilsequences and bracket polynomials) as an assertion of the pseudorandomness of the Liouville function from a computational complexity perspective. Suppose you possess a calculator with the wonderful property of being infinite precision: it can accept arbitrarily large real numbers as input, manipulate them precisely, and also store them in memory. However, this calculator has two limitations. Firstly, the only operations available are addition, subtraction, multiplication, integer part , fractional part
, memory store (into one of O(1) registers), and memory recall (from one of these O(1) registers). In particular, there is no ability to perform division. Secondly, the calculator only has a finite display screen, and when it shows a real number, it only shows O(1) digits before and after the decimal point. (Thus, for instance, the real number 1234.56789 might be displayed only as
.)
Now suppose you play the following game with an opponent.
- The opponent specifies a large integer d.
- You get to enter in O(1) real constants of your choice into your calculator. These can be absolute constants such as
and
, or they can depend on d (e.g. you can enter in
).
- The opponent randomly selects an d-digit integer n, and enters n into one of the registers of your calculator.
- You are allowed to perform O(1) operations on your calculator and record what is displayed on the calculator’s viewscreen.
- After this, you have to guess whether the opponent’s number n had an odd or even number of prime factors (i.e. you guess
.)
- If you guess correctly, you win $1; otherwise, you lose $1.
For instance, using your calculator you can work out the first few digits of , provided of course that you entered the constants
and
in advance. You can also work out the leading digits of n by storing
in advance, and computing the first few digits of
.
Our theorem is equivalent to the assertion that as d goes to infinity (keeping the O(1) constants fixed), your probability of winning this game converges to 1/2; in other words, your calculator becomes asymptotically useless to you for the purposes of guessing whether n has an odd or even number of prime factors, and you may as well just guess randomly.
[I should mention a recent result in a similar spirit by Mauduit and Rivat; in this language, their result asserts that knowing the last few digits of the digit-sum of n does not increase your odds of guessing correctly.]
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