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A sequence {a: {\bf Z} \rightarrow {\bf C}} of complex numbers is said to be quasiperiodic if it is of the form

\displaystyle a(n) = F( \alpha_1 n \hbox{ mod } 1, \dots, \alpha_k n \hbox{ mod } 1)

for some real numbers {\alpha_1,\dots,\alpha_k} and continuous function {F: ({\bf R}/{\bf Z})^k \rightarrow {\bf C}}. For instance, linear phases such as {n \mapsto e(\alpha n + \beta)} (where {e(\theta) := e^{2\pi i \theta}}) are examples of quasiperiodic sequences; the top order coefficient {\alpha} (modulo {1}) can be viewed as a “frequency” of the integers, and an element of the Pontryagin dual {\hat {\bf Z} \equiv {\bf R}/{\bf Z}} of the integers. Any periodic sequence is also quasiperiodic (taking {k=1} and {\alpha_1} 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

\displaystyle a(n) = \sum_{j=1}^\infty c_j e(\alpha_j n)

with {\alpha_1,\alpha_2,\dots} real numbers and {c_1,c_2,\dots} 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 {(X, \mu, T)} is a measure-preserving system – a probability space {(X,\mu)} equipped with a measure-preserving shift, and {f_1,f_2 \in L^\infty(X)} are bounded measurable functions, then the correlation sequence

\displaystyle a(n) := \int_X f_1(x) f_2(T^n x)\ d\mu(x)

can be shown to be an almost periodic sequence, plus an error term {b_n} which is “null” in the sense that it has vanishing uniform density:

\displaystyle \sup_N \frac{1}{M} \sum_{n=N+1}^{N+M} |b_n| \rightarrow 0 \hbox{ as } M \rightarrow \infty.

This can be established in a number of ways, for instance by writing {a(n)} as the Fourier coefficients of the spectral measure of the shift {T} with respect to the functions {f_1,f_2}, 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 {\alpha n + \beta} are replaced with higher degree counterparts. The most obvious candidates for these counterparts would be the polynomials {\alpha_d n^d + \dots + \alpha_0}, 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 {{\bf Z}}). 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 {{\bf C}^m} rather than in {{\bf C}}. 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 {{\bf C}^m}, and for sequences taking values in different vector spaces {{\bf C}^m}, {{\bf C}^{m'}}, we may utilise the tensor product {\otimes: {\bf C}^m \times {\bf C}^{m'} \rightarrow {\bf C}^{mm'}}, which we will normalise by defining

\displaystyle (z_1,\dots,z_m) \otimes (w_1,\dots,w_{m'}) = (z_1 w_1, \dots, z_1 w_{m'}, \dots, z_m w_1, \dots, z_m w_{m'} ).

This product is associative and bilinear, and also commutative up to permutation of the indices. It also interacts well with the Hermitian norm

\displaystyle \| (z_1,\dots,z_m) \| := \sqrt{|z_1|^2 + \dots + |z_m|^2}

since we have {\|z \otimes w\| = \|z\| \|w\|}.

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 {d} is a quotient {G/\Gamma}, where {G} is a connected, simply connected nilpotent Lie group of step at most {d} (thus, all {d+1}-fold commutators vanish) and {\Gamma} is a discrete cocompact lattice in {G}. A basic nilsequence of degree at most {d} is a sequence of the form {n \mapsto F(g^n g_0 \Gamma)}, where {g_0 \Gamma \in G/\Gamma}, {g \in G}, and {F: G/\Gamma \rightarrow {\bf C}^m} 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 {1} if and only if it is quasiperiodic. The requirement that {G} 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 {\{ N+1, \dots, N+M\}}), it is common to impose additional regularity conditions on the function {F}, 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 {d \geq 0}. A filtered group of degree at most {d} is a group {G} together with a sequence {G_\bullet = (G_0,G_1,G_2,\dots)} of subgroups {G \geq G_0 \geq G_1 \geq \dots} with {G_{d+1}=\{\hbox{id}\}} and {[G_i,G_j] \subset G_{i+j}} for {i,j \geq 0}. A polynomial sequence {g: {\bf Z} \rightarrow G} into a filtered group is a function such that {\partial_{h_i} \dots \partial_{h_1} g(n) \in G_i} for all {i \geq 0} and {n,h_1,\dots,h_i \in{\bf Z}}, where {\partial_h g(n) := g(n+h) g(n)^{-1}} is the difference operator. A filtered nilmanifold of degree at most {s} is a quotient {G/\Gamma}, where {G} is a filtered group of degree at most {s} such that {G} and all of the subgroups {G_i} are connected, simply connected nilpotent filtered Lie group, and {\Gamma} is a discrete cocompact subgroup of {G} such that {\Gamma_i := \Gamma \cap G_i} is a discrete cocompact subgroup of {G_i}. A basic nilsequence of degree at most {d} is a sequence of the form {n \mapsto F(g(n))}, where {g: {\bf Z} \rightarrow G} is a polynomial sequence, {G/\Gamma} is a filtered nilmanifold of degree at most {d}, and {F: G \rightarrow {\bf C}^m} is a continuous function which is {\Gamma}-automorphic, in the sense that {F(g \gamma) = F(g)} for all {g \in G} and {\gamma \in \Gamma}.

One can easily identify a {\Gamma}-automorphic function on {G} with a function on {G/\Gamma}, but there are some (very minor) advantages to working on the group {G} instead of the quotient {G/\Gamma}, as it becomes slightly easier to modify the automorphy group {\Gamma} when needed. (But because the action of {\Gamma} on {G} is free, one can pass from {\Gamma}-automorphic functions on {G} to functions on {G/\Gamma} with very little difficulty.) The main reason to work with polynomial sequences {n \mapsto g(n)} rather than geometric progressions {n \mapsto g^n g_0 \Gamma} 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 {d} in the sense of the first definition, is also a basic nilsequence of degree at most {d} in the second definition, since a nilmanifold of degree at most {d} can be filtered using the lower central series, and any linear sequence {n \mapsto g^n g_0} 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

\displaystyle a(n) = F( P_1(n), \dots, P_k(n) ),

where {P_1,\dots,P_k: {\bf Z} \rightarrow {\bf R}} are polynomials of degree at most {d}, and {F: {\bf R}^k \rightarrow {\bf C}^m} is a {{\bf Z}^k}-automorphic (i.e., {{\bf Z}^k}-periodic) continuous function. The map {P: {\bf Z} \rightarrow {\bf R}^k} defined by {P(n) := (P_1(n),\dots,P_k(n))} is a polynomial map of degree at most {d}, if one filters {{\bf R}^k} by defining {({\bf R}^k)_i} to equal {{\bf R}^k} when {i \leq d}, and {\{0\}} for {i > d}. The torus {{\bf R}^k/{\bf Z}^k} then becomes a filtered nilmanifold of degree at most {d}, and {a(n)} is thus a basic nilsequence of degree at most {d} as per the second definition. It is also possible explicitly describe {a_n} as a basic nilsequence of degree at most {d} as per the first definition, for instance (in the {k=1} case) by taking {G} to be the space of upper triangular unipotent {d+1 \times d+1} real matrices, and {\Gamma} the subgroup with integer coefficients; we leave the details to the interested reader.

The other key example is a sequence of the form

\displaystyle a(n) = F( \alpha n, \{ \alpha n \} \beta n )

where {\alpha,\beta} are real numbers, {\{ \alpha n \} = \alpha n - \lfloor \alpha n \rfloor} denotes the fractional part of {\alpha n}, and and {F: {\bf R}^2 \rightarrow {\bf C}^m} is a {{\bf Z}^2}-automorphic continuous function that vanishes in a neighbourhood of {{\bf Z} \times {\bf R}}. To describe this as a nilsequence, we use the nilpotent connected, simply connected degree {2}, Heisenberg group

\displaystyle G := \begin{pmatrix} 1 & {\bf R} & {\bf R} \\ 0 & 1 & {\bf R} \\ 0 & 0 & 1 \end{pmatrix}

with the lower central series filtration {G_0=G_1=G}, {G_2= [G,G] = \begin{pmatrix} 1 &0 & {\bf R} \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{pmatrix}}, and {G_i = \{ \mathrm{id} \}} for {i > 2}, {\Gamma} to be the discrete compact subgroup

\displaystyle \Gamma := \begin{pmatrix} 1 & {\bf Z} & {\bf Z} \\ 0 & 1 & {\bf Z} \\ 0 & 0 & 1 \end{pmatrix},

{g: {\bf Z} \rightarrow G} to be the polynomial sequence

\displaystyle g(n) := \begin{pmatrix} 1 & \beta n & \alpha \beta n^2 \\ 0 & 1 & \alpha n \\ 0 & 0 & 1 \end{pmatrix}

and {\tilde F: G \rightarrow {\bf C}^m} to be the {\Gamma}-automorphic function

\displaystyle \tilde F( \begin{pmatrix} 1 & x & y \\ 0 & 1 & z \\ 0 & 0 & 1 \end{pmatrix} ) = F( \{ z \}, y - \lfloor z \rfloor x );

one easily verifies that this function is indeed {\Gamma}-automorphic, and it is continuous thanks to the vanishing properties of {F}. Also we have {a(n) = \tilde F(g(n))}, so {a} is a basic nilsequence of degree at most {2}. One can concoct similar examples with {\{ \alpha n \} \beta n} replaced by other “bracket polynomials” of {n}; for instance

\displaystyle a(n) = F( \alpha n, \{ \alpha n - \frac{1}{2} \} \beta n )

will be a basic nilsequence if {F} now vanishes in a neighbourhood of {(\frac{1}{2}+{\bf Z}) \times {\bf R}} rather than {{\bf Z} \times {\bf R}}. 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 {d} is defined to be a sequence that is the uniform limit of basic nilsequences of degree at most {d}. Thus for instance a sequence is a nilsequence of degree at most {1} if and only if it is almost periodic, while a sequence is a nilsequence of degree at most {0} if and only if it is constant. Such objects arise in higher order recurrence: for instance, if {h_0,\dots,h_d} are integers, {(X,\mu,T)} is a measure-preserving system, and {f_0,\dots,f_d \in L^\infty(X)}, then it was shown by Leibman that the sequence

\displaystyle n \mapsto \int_X f_0(T^{h_0 n} x) \dots f_d(T^{h_d n} x)\ d\mu(x)

is equal to a nilsequence of degree at most {d}, plus a null sequence. (The special case when the measure-preserving system was ergodic and {h_i = i} for {i=0,\dots,d} 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 {a: {\bf Z} \rightarrow {\bf C}^m} is a basic nilsequence of degree at most {d} if and only if each of its {m} components are. The scalar basic nilsequences {a: {\bf Z} \rightarrow {\bf C}} of degree {d} 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 {a: {\bf Z} \rightarrow {\bf C}^m} of degree at most {d} form a complex vector space closed under complex conjugation for each {m}, and that the tensor product of any two basic nilsequences of degree at most {d} is another basic nilsequence of degree at most {d}. 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 {d \geq 1}. A sub-nilcharacter of degree {d} is a basic nilsequence {\chi: n \mapsto F(g(n))} of degree at most {d}, such that {F} obeys the additional modulation property

\displaystyle F( g_d g ) = e( \xi \cdot g_d ) F(g) \ \ \ \ \ (1)

for all {g \in G} and {g_d \in G_d}, where {\xi: G_d \rightarrow {\bf R}} is a continuous homomorphism {g_d \mapsto \xi \cdot g_d}. (Note from (1) and {\Gamma}-automorphy that unless {F} vanishes identically, {\xi} must map {\Gamma_d} to {{\bf Z}}, thus without loss of generality one can view {\xi} as an element of the Pontryagial dual of the torus {G_d / \Gamma_d}.) If in addition one has {\|F(g)\|=1} for all {g \in G}, we call {\chi} a nilcharacter of degree {d \geq 1}.

In the degree one case {d=1}, the only sub-nilcharacters are of the form {\chi(n) = e(\alpha n)} for some vector {c \in {\bf C}^m} and {\alpha \in {\bf R}}, and this is a nilcharacter if {c} is a unit vector. Similarly, in higher degree, any sequence of the form {\chi(n) = c e(P(n))}, where {c \in {\bf C}^m} is a vector and {P: {\bf Z} \rightarrow {\bf R}} is a polynomial of degree at most {d}, is a sub-nilcharacter of degree {d}, and a character if {c} is a unit vector. A nilsequence of degree at most {d-1} is automatically a sub-nilcharacter of degree {d}, and a nilcharacter if it is of magnitude {1}. A further example of a nilcharacter is provided by the two-dimensional sequence {\chi: {\bf Z} \rightarrow {\bf C}^2} defined by

\displaystyle \chi(n) := ( F_0( \alpha n ) e( \{ \alpha n \} \beta n ), F_{1/2}( \alpha n ) e( \{ \alpha n - \frac{1}{2} \} \beta n ) ) \ \ \ \ \ (2)

where {F_0, F_{1/2}: {\bf R} \rightarrow {\bf C}} are continuous, {{\bf Z}}-automorphic functions that vanish on a neighbourhood of {{\bf Z}} and {\frac{1}{2}+{\bf Z}} respectively, and which form a partition of unity in the sense that

\displaystyle |F_0(x)|^2 + |F_{1/2}(x)|^2 = 1

for all {x \in {\bf R}}. Note that one needs both {F_0} and {F_{1/2}} 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 {d} sub-nilcharacter {f: {\bf Z} \rightarrow {\bf C}^m} can be expressed in the form {f = c \chi}, where {\chi: {\bf Z} \rightarrow {\bf C}^{m'}} is a degree {d} nilcharacter, and {c: {\bf C}^{m'} \rightarrow {\bf C}^m} is a linear transformation. Indeed, by scaling we may assume {f(n) = F(g(n))} where {|F| < 1} uniformly. Using partitions of unity, one can find further functions {F_1,\dots,F_m} also obeying (1) for the same character {\xi} such that {|F_1|^2 + \dots + |F_m|^2} is non-zero; by dividing out the {F_1,\dots,F_m} by the square root of this quantity, and then multiplying by {\sqrt{1-|F|^2}}, we may assume that

\displaystyle |F|^2 + |F_1|^2 + \dots + |F_m|^2 = 1,

and then

\displaystyle \chi(n) := (F(g(n)), F_1(g(n)), \dots, F_m(g(n)))

becomes a degree {d} nilcharacter that contains {f(n)} 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 {d} nilcharacters forms a semigroup under tensor product, with the constant sequence {1} as the identity. One can upgrade this semigroup to an abelian group by quotienting nilcharacters out by equivalence:

Definition 4 Let {d \geq 1}. We say that two degree {d} nilcharacters {\chi: {\bf Z} \rightarrow {\bf C}^m}, {\chi': {\bf Z} \rightarrow {\bf C}^{m'}} are equivalent if {\chi \otimes \overline{\chi'}: {\bf Z} \rightarrow {\bf C}^{mm'}} is equal (as a sequence) to a basic nilsequence of degree at most {d-1}. (We will later show that this is indeed an equivalence relation.) The equivalence class {[\chi]_{\mathrm{Symb}^d({\bf Z})}} 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 {\mathrm{Symb}^d({\bf Z})}.

As we shall see below the fold, {\mathrm{Symb}^d({\bf Z})} 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 {d=1}, the group is isomorphic to the Ponytragin dual {\hat {\bf Z} = {\bf R}/{\bf Z}} of the integers, and {\mathrm{Symb}^d({\bf Z})} for {d > 1} should be viewed as higher order generalisations of this Pontryagin dual. In principle, this group can be explicitly described for all {d}, but the theory rapidly gets complicated as {d} increases (much as the classification of nilpotent Lie groups or Lie algebras of step {d} rapidly gets complicated even for medium-sized {d} such as {d=3} or {d=4}). We will give an explicit description of the {d=2} case here. There is however one nice (and non-trivial) feature of {\mathrm{Symb}^d({\bf Z})} for {d \geq 2} – it is not just an abelian group, but is in fact a vector space over the rationals {{\bf Q}}!

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The equidistribution theorem asserts that if {\alpha \in {\bf R}/{\bf Z}} is an irrational phase, then the sequence {(n\alpha)_{n=1}^\infty} is equidistributed on the unit circle, or equivalently that

\displaystyle \frac{1}{N} \sum_{n=1}^N F(n\alpha) \rightarrow \int_{{\bf R}/{\bf Z}} F(x)\ dx

for any continuous (or equivalently, for any smooth) function {F: {\bf R}/{\bf Z} \rightarrow {\bf C}}. By approximating {F} uniformly by a Fourier series, this claim is equivalent to that of showing that

\displaystyle \frac{1}{N} \sum_{n=1}^N e(hn\alpha) \rightarrow 0

for any non-zero integer {h} (where {e(x) := e^{2\pi i x}}), which is easily verified from the irrationality of {\alpha} and the geometric series formula. Conversely, if {\alpha} is rational, then clearly {\frac{1}{N} \sum_{n=1}^N e(hn\alpha)} fails to go to zero when {h} is a multiple of the denominator of {\alpha}.

One can then ask for more quantitative information about the decay of exponential sums of {\frac{1}{N} \sum_{n=1}^N e(n \alpha)}, or more generally on exponential sums of the form {\frac{1}{|Q|} \sum_{n \in Q} e(P(n))} for an arithmetic progression {Q} (in this post all progressions are understood to be finite) and a polynomial {P: Q \rightarrow \/{\bf Z}}. It will be convenient to phrase such information in the form of an inverse theorem, describing those phases for which the exponential sum is large. Indeed, we have

Lemma 1 (Geometric series formula, inverse form) Let {Q \subset {\bf Z}} be an arithmetic progression of length at most {N} for some {N \geq 1}, and let {P(n) = n \alpha + \beta} be a linear polynomial for some {\alpha,\beta \in {\bf R}/{\bf Z}}. If

\displaystyle \frac{1}{N} |\sum_{n \in Q} e(P(n))| \geq \delta

for some {\delta > 0}, then there exists a subprogression {Q'} of {Q} of size {|Q'| \gg \delta^2 N} such that {P(n)} varies by at most {\delta} on {Q'} (that is to say, {P(n)} lies in a subinterval of {{\bf R}/{\bf Z}} of length at most {\delta}).

Proof: By a linear change of variable we may assume that {Q} is of the form {\{0,\dots,N'-1\}} for some {N' \geq 1}. We may of course assume that {\alpha} is non-zero in {{\bf R}/{\bf Z}}, so that {\|\alpha\|_{{\bf R}/{\bf Z}} > 0} ({\|x\|_{{\bf R}/{\bf Z}}} denotes the distance from {x} to the nearest integer). From the geometric series formula we see that

\displaystyle |\sum_{n \in Q} e(P(n))| \leq \frac{2}{|e(\alpha) - 1|} \ll \frac{1}{\|\alpha\|_{{\bf R}/{\bf Z}}},

and so {\|\alpha\|_{{\bf R}/{\bf Z}} \ll \frac{1}{\delta N}}. Setting {Q' := \{ n \in Q: n \leq c \delta^2 N \}} for some sufficiently small absolute constant {c}, we obtain the claim. \Box

Thus, in order for a linear phase {P(n)} to fail to be equidistributed on some long progression {Q}, {P} must in fact be almost constant on large piece of {Q}.

As is well known, this phenomenon generalises to higher order polynomials. To achieve this, we need two elementary additional lemmas. The first relates the exponential sums of {P} to the exponential sums of its “first derivatives” {n \mapsto P(n+h)-P(n)}.

Lemma 2 (Van der Corput lemma, inverse form) Let {Q \subset {\bf Z}} be an arithmetic progression of length at most {N}, and let {P: Q \rightarrow {\bf R}/{\bf Z}} be an arbitrary function such that

\displaystyle \frac{1}{N} |\sum_{n \in Q} e(P(n))| \geq \delta \ \ \ \ \ (1)

 

for some {\delta > 0}. Then, for {\gg \delta^2 N} integers {h \in Q-Q}, there exists a subprogression {Q_h} of {Q}, of the same spacing as {Q}, such that

\displaystyle \frac{1}{N} |\sum_{n \in Q_h} e(P(n+h)-P(n))| \gg \delta^2. \ \ \ \ \ (2)

 

Proof: Squaring (1), we see that

\displaystyle \sum_{n,n' \in Q} e(P(n') - P(n)) \geq \delta^2 N^2.

We write {n' = n+h} and conclude that

\displaystyle \sum_{h \in Q-Q} \sum_{n \in Q_h} e( P(n+h)-P(n) ) \geq \delta^2 N^2

where {Q_h := Q \cap (Q-h)} is a subprogression of {Q} of the same spacing. Since {\sum_{n \in Q_h} e( P(n+h)-P(n) ) = O(N)}, we conclude that

\displaystyle |\sum_{n \in Q_h} e( P(n+h)-P(n) )| \gg \delta^2 N

for {\gg \delta^2 N} values of {h} (this can be seen, much like the pigeonhole principle, by arguing via contradiction for a suitable choice of implied constants). The claim follows. \Box

The second lemma (which we recycle from this previous blog post) is a variant of the equidistribution theorem.

Lemma 3 (Vinogradov lemma) Let {I \subset [-N,N] \cap {\bf Z}} be an interval for some {N \geq 1}, and let {\theta \in{\bf R}/{\bf Z}} be such that {\|n\theta\|_{{\bf R}/{\bf Z}} \leq \varepsilon} for at least {\delta N} values of {n \in I}, for some {0 < \varepsilon, \delta < 1}. Then either

\displaystyle N < \frac{2}{\delta}

or

\displaystyle \varepsilon > 10^{-2} \delta

or else there is a natural number {q \leq 2/\delta} such that

\displaystyle \| q \theta \|_{{\bf R}/{\bf Z}} \ll \frac{\varepsilon}{\delta N}.

Proof: We may assume that {N \geq \frac{2}{\delta}} and {\varepsilon \leq 10^{-2} \delta}, since we are done otherwise. Then there are at least two {n \in I} with {\|n \theta \|_{{\bf R}/{\bf Z}} \leq \varepsilon}, and by the pigeonhole principle we can find {n_1 < n_2} in {Q} with {\|n_1 \theta \|_{{\bf R}/{\bf Z}}, \|n_2 \theta \|_{{\bf R}/{\bf Z}} \leq \varepsilon} and {n_2-n_1 \leq \frac{2}{\delta}}. By the triangle inequality, we conclude that there exists at least one natural number {q \leq \frac{2}{\delta}} for which

\displaystyle \| q \theta \|_{{\bf R}/{\bf Z}} \leq 2\varepsilon.

We take {q} to be minimal amongst all such natural numbers, then we see that there exists {a} coprime to {q} and {|\kappa| \leq 2\varepsilon} such that

\displaystyle \theta = \frac{a}{q} + \frac{\kappa}{q}. \ \ \ \ \ (3)

 

If {\kappa=0} then we are done, so suppose that {\kappa \neq 0}. Suppose that {n < m} are elements of {I} such that {\|n\theta \|_{{\bf R}/{\bf Z}}, \|m\theta \|_{{\bf R}/{\bf Z}} \leq \varepsilon} and {m-n \leq \frac{1}{10 \kappa}}. Writing {m-n = qk + r} for some {0 \leq r < q}, we have

\displaystyle \| (m-n) \theta \|_{{\bf R}/{\bf Z}} = \| \frac{ra}{q} + (m-n) \frac{\kappa}{q} \|_{{\bf R}/{\bf Z}} \leq 2\varepsilon.

By hypothesis, {(m-n) \frac{\kappa}{q} \leq \frac{1}{10 q}}; note that as {q \leq 2/\delta} and {\varepsilon \leq 10^{-2} \delta} we also have {\varepsilon \leq \frac{1}{10q}}. This implies that {\| \frac{ra}{q} \|_{{\bf R}/{\bf Z}} < \frac{1}{q}} and thus {r=0}. We then have

\displaystyle |k \kappa| \leq 2 \varepsilon.

We conclude that for fixed {n \in I} with {\|n\theta \|_{{\bf R}/{\bf Z}} \leq \varepsilon}, there are at most {\frac{2\varepsilon}{|\kappa|}} elements {m} of {[n, n + \frac{1}{10 |\kappa|}]} such that {\|m\theta \|_{{\bf R}/{\bf Z}} \leq \varepsilon}. Iterating this with a greedy algorithm, we see that the number of {n \in I} with {\|n\theta \|_{{\bf R}/{\bf Z}} \leq \varepsilon} is at most {(\frac{N}{1/10|\kappa|} + 1) 2\varepsilon/|\kappa|}; since {\varepsilon < 10^{-2} \delta}, this implies that

\displaystyle \delta N \ll 2 \varepsilon / \kappa

and the claim follows. \Box

Now we can quickly obtain a higher degree version of Lemma 1:

Proposition 4 (Weyl exponential sum estimate, inverse form) Let {Q \subset {\bf Z}} be an arithmetic progression of length at most {N} for some {N \geq 1}, and let {P: {\bf Z} \rightarrow {\bf R}/{\bf Z}} be a polynomial of some degree at most {d \geq 0}. If

\displaystyle \frac{1}{N} |\sum_{n \in Q} e(P(n))| \geq \delta

for some {\delta > 0}, then there exists a subprogression {Q'} of {Q} with {|Q'| \gg_d \delta^{O_d(1)} N} such that {P} varies by at most {\delta} on {Q'}.

Proof: We induct on {d}. The cases {d=0,1} are immediate from Lemma 1. Now suppose that {d \geq 2}, and that the claim had already been proven for {d-1}. To simplify the notation we allow implied constants to depend on {d}. Let the hypotheses be as in the proposition. Clearly {\delta} cannot exceed {1}. By shrinking {\delta} as necessary we may assume that {\delta \leq c} for some sufficiently small constant {c} depending on {d}.

By rescaling we may assume {Q \subset [0,N] \cap {\bf Z}}. By Lemma 3, we see that for {\gg \delta^2 N} choices of {h \in [-N,N] \cap {\bf Z}} such that

\displaystyle \frac{1}{N} |\sum_{n \in I_h} e(P(n+h) - P(n))| \gg \delta^2

for some interval {I_h \subset [0,N] \cap {\bf Z}}. We write {P(n) = \sum_{i \leq d} \alpha_i n^i}, then {P(n+h)-P(n)} is a polynomial of degree at most {d-1} with leading coefficient {h \alpha_d n^{d-1}}. We conclude from induction hypothesis that for each such {h}, there exists a natural number {q_h \ll \delta^{-O(1)}} such that {\|q_h h \alpha_d \|_{{\bf R}/{\bf Z}} \ll \delta^{-O(1)} / N^{d-1}}, by double-counting, this implies that there are {\gg \delta^{O(1)} N} integers {n} in the interval {[-\delta^{-O(1)} N, \delta^{-O(1)} N] \cap {\bf Z}} such that {\|n \alpha_d \|_{{\bf R}/{\bf Z}} \ll \delta^{-O(1)} / N^{d-1}}. Applying Lemma 3, we conclude that either {N \ll \delta^{-O(1)}}, or that

\displaystyle \| q \alpha_d \|_{{\bf R}/{\bf Z}} \ll \delta^{-O(1)} / N^d. \ \ \ \ \ (4)

 

In the former case the claim is trivial (just take {Q'} to be a point), so we may assume that we are in the latter case.

We partition {Q} into arithmetic progressions {Q'} of spacing {q} and length comparable to {\delta^{-C} N} for some large {C} depending on {d} to be chosen later. By hypothesis, we have

\displaystyle \frac{1}{|Q|} |\sum_{n \in Q} e(P(n))| \geq \delta

so by the pigeonhole principle, we have

\displaystyle \frac{1}{|Q'|} |\sum_{n \in Q'} e(P(n))| \geq \delta

for at least one such progression {Q'}. On this progression, we may use the binomial theorem and (4) to write {\alpha_d n^d} as a polynomial in {n} of degree at most {d-1}, plus an error of size {O(\delta^{C - O(1)})}. We thus can write {P(n) = P'(n) + O(\delta^{C-O(1)})} for {n \in Q'} for some polynomial {P'} of degree at most {d-1}. By the triangle inequality, we thus have (for {C} large enough) that

\displaystyle \frac{1}{|Q'|} |\sum_{n \in Q'} e(P'(n))| \gg \delta

and hence by induction hypothesis we may find a subprogression {Q''} of {Q'} of size {|Q''| \gg \delta^{O(1)} N} such that {P'} varies by most {\delta/2} on {Q''}, and thus (for {C} large enough again) that {P} varies by at most {\delta} on {Q''}, and the claim follows. \Box

This gives the following corollary (also given as Exercise 16 in this previous blog post):

Corollary 5 (Weyl exponential sum estimate, inverse form II) Let {I \subset [-N,N] \cap {\bf Z}} be a discrete interval for some {N \geq 1}, and let {P(n) = \sum_{i \leq d} \alpha_i n^i} polynomial of some degree at most {d \geq 0} for some {\alpha_0,\dots,\alpha_d \in {\bf R}/{\bf Z}}. If

\displaystyle \frac{1}{N} |\sum_{n \in I} e(P(n))| \geq \delta

for some {\delta > 0}, then there is a natural number {q \ll_d \delta^{-O_d(1)}} such that {\| q\alpha_i \|_{{\bf R}/{\bf Z}} \ll_d \delta^{-O_d(1)} N^{-i}} for all {i=0,\dots,d}.

One can obtain much better exponents here using Vinogradov’s mean value theorem; see Theorem 1.6 this paper of Wooley. (Thanks to Mariusz Mirek for this reference.) However, this weaker result already suffices for many applications, and does not need any result as deep as the mean value theorem.

Proof: To simplify notation we allow implied constants to depend on {d}. As before, we may assume that {\delta \leq c} for some small constant {c>0} depending only on {d}. We may also assume that {N \geq \delta^{-C}} for some large {C}, as the claim is trivial otherwise (set {q=1}).

Applying Proposition 4, we can find a natural number {q \ll \delta^{-O(1)}} and an arithmetic subprogression {Q} of {I} such that {|Q| \gg \delta^{O(1)}} and such that {P} varies by at most {\delta} on {Q}. Writing {Q = \{ qn+r: n \in I'\}} for some interval {I' \subset [0,N] \cap {\bf Z}} of length {\gg \delta^{O(1)}} and some {0 \leq r < q}, we conclude that the polynomial {n \mapsto P(qn+r)} varies by at most {\delta} on {I'}. Taking {d^{th}} order differences, we conclude that the {d^{th}} coefficient of this polynomial is {O(\delta^{-O(1)} / N^d)}; by the binomial theorem, this implies that {n \mapsto P(qn+r)} differs by at most {O(\delta)} on {I'} from a polynomial of degree at most {d-1}. Iterating this, we conclude that the {i^{th}} coefficient of {n \mapsto P(qn+r)} is {O(\delta N^{-i})} for {i=0,\dots,d}, and the claim then follows by inverting the change of variables {n \mapsto qn+r} (and replacing {q} with a larger quantity such as {q^d} as necessary). \Box

For future reference we also record a higher degree version of the Vinogradov lemma.

Lemma 6 (Polynomial Vinogradov lemma) Let {I \subset [-N,N] \cap {\bf Z}} be a discrete interval for some {N \geq 1}, and let {P: {\bf Z} \rightarrow {\bf R}/{\bf Z}} be a polynomial {P(n) = \sum_{i \leq d} \alpha_i n^i} of degree at most {d} for some {d \geq 1} such that {\|P(n)\|_{{\bf R}/{\bf Z}} \leq \varepsilon} for at least {\delta N} values of {n \in I}, for some {0 < \varepsilon, \delta < 1}. Then either

\displaystyle N \ll_d \delta^{-O_d(1)} \ \ \ \ \ (5)

 

or

\displaystyle \varepsilon \gg_d \delta^{O_d(1)} \ \ \ \ \ (6)

 

or else there is a natural number {q \ll_d \delta^{-O_d(1)}} such that

\displaystyle \| q \alpha_i \|_{{\bf R}/{\bf Z}} \ll \frac{\delta^{-O(1)} \varepsilon}{N^i}

for all {i=0,\dots,d}.

Proof: We induct on {d}. For {d=1} this follows from Lemma 3 (noting that if {\|P(n)\|_{{\bf R}/{\bf Z}}, \|P(n_0)\|_{{\bf R}/Z} \leq \varepsilon} then {\|P(n)-P(n_0)\|_{{\bf R}/{\bf Z}} \leq 2\varepsilon}), so suppose that {d \geq 2} and that the claim is already proven for {d-1}. We now allow all implied constants to depend on {d}.

For each {h \in [-2N,2N] \cap {\bf Z}}, let {N_h} denote the number of {n \in [-N,N] \cap {\bf Z}} such that {\| P(n+h)\|_{{\bf R}/{\bf Z}}, \|P(n)\|_{{\bf R}/{\bf Z}} \leq \varepsilon}. By hypothesis, {\sum_{h \in [-2N,2N] \cap {\bf Z}} N_h \gg \delta^2 N^2}, and clearly {N_h = O(N)}, so we must have {N_h \gg \delta^2 N} for {\gg \delta^2 N} choices of {h}. For each such {h}, we then have {\|P(n+h)-P(n)\|_{{\bf R}/{\bf Z}} \leq 2\varepsilon} for {\gg \delta^2 N} choices of {n \in [-N,N] \cap {\bf Z}}, so by induction hypothesis, either (5) or (6) holds, or else for {\gg \delta^{O(1)} N} choices of {h \in [-2N,2N] \cap {\bf Z}}, there is a natural number {q_h \ll \delta^{-O(1)}} such that

\displaystyle \| q_h \alpha_{i,h} \|_{{\bf R}/{\bf Z}} \ll \frac{\delta^{-O(1)} \varepsilon}{N^i}

for {i=1,\dots,d-1}, where {\alpha_{i,h}} are the coefficients of the degree {d-1} polynomial {n \mapsto P(n+h)-P(n)}. We may of course assume it is the latter which holds. By the pigeonhole principle we may take {q_h= q} to be independent of {h}.

Since {\alpha_{d-1,h} = dh \alpha_d}, we have

\displaystyle \| qd h \alpha_d \|_{{\bf R}/{\bf Z}} \ll \frac{\delta^{-O(1)} \varepsilon}{N^{d-1}}

for {\gg \delta^{O(1)} N} choices of {h}, so by Lemma 3, either (5) or (6) holds, or else (after increasing {q} as necessary) we have

\displaystyle \| q \alpha_d \|_{{\bf R}/{\bf Z}} \ll \frac{\delta^{-O(1)} \varepsilon}{N^d}.

We can again assume it is the latter that holds. This implies that {q \alpha_{d-2,h} = (d-1) h \alpha_{d-1} + O( \delta^{-O(1)} \varepsilon / N^{d-2} )} modulo {1}, so that

\displaystyle \| q(d-1) h \alpha_{d-1} \|_{{\bf R}/{\bf Z}} \ll \frac{\delta^{-O(1)} \varepsilon}{N^{d-2}}

for {\gg \delta^{O(1)} N} choices of {h}. Arguing as before and iterating, we obtain the claim. \Box

The above results also extend to higher dimensions. Here is the higher dimensional version of Proposition 4:

Proposition 7 (Multidimensional Weyl exponential sum estimate, inverse form) Let {k \geq 1} and {N_1,\dots,N_k \geq 1}, and let {Q_i \subset {\bf Z}} be arithmetic progressions of length at most {N_i} for each {i=1,\dots,k}. Let {P: {\bf Z}^k \rightarrow {\bf R}/{\bf Z}} be a polynomial of degrees at most {d_1,\dots,d_k} in each of the {k} variables {n_1,\dots,n_k} separately. If

\displaystyle \frac{1}{N_1 \dots N_k} |\sum_{n \in Q_1 \times \dots \times Q_k} e(P(n))| \geq \delta

for some {\delta > 0}, then there exists a subprogression {Q'_i} of {Q_i} with {|Q'_i| \gg_{k,d_1,\dots,d_k} \delta^{O_{k,d_1,\dots,d_k}(1)} N_i} for each {i=1,\dots,k} such that {P} varies by at most {\delta} on {Q'_1 \times \dots \times Q'_k}.

A much more general statement, in which the polynomial phase {n \mapsto e(P(n))} is replaced by a nilsequence, and in which one does not necessarily assume the exponential sum is small, is given in Theorem 8.6 of this paper of Ben Green and myself, but it involves far more notation to even state properly.

Proof: We induct on {k}. The case {k=1} was established in Proposition 5, so we assume that {k \geq 2} and that the claim has already been proven for {k-1}. To simplify notation we allow all implied constants to depend on {k,d_1,\dots,d_k}. We may assume that {\delta \leq c} for some small {c>0} depending only on {k,d_1,\dots,d_k}.

By a linear change of variables, we may assume that {Q_i \subset [0,N_i] \cap {\bf Z}} for all {i=1,\dots,k}.

We write {n' := (n_1,\dots,n_{k-1})}. First suppose that {N_k = O(\delta^{-O(1)})}. Then by the pigeonhole principle we can find {n_k \in I_k} such that

\displaystyle \frac{1}{N_1 \dots N_{k-1}} |\sum_{n' \in Q_1 \times \dots \times Q_{k-1}} e(P(n',n_k))| \geq \delta

and the claim then follows from the induction hypothesis. Thus we may assume that {N_k \geq \delta^{-C}} for some large {C} depending only on {k,d_1,\dots,d_k}. Similarly we may assume that {N_i \geq \delta^{-C}} for all {i=1,\dots,k}.

By the triangle inequality, we have

\displaystyle \frac{1}{N_1 \dots N_k} \sum_{n_k \in Q_k} |\sum_{n' \in Q_1 \times \dots \times Q_{k-1}} e(P(n',n_k))| \geq \delta.

The inner sum is {O(N_k)}, and the outer sum has {O(N_1 \dots N_{k-1})} terms. Thus, for {\gg \delta N_1 \dots N_{k-1}} choices of {n' \in Q_1 \times \dots \times Q_{k-1}}, one has

\displaystyle \frac{1}{N_k} |\sum_{n_k \in Q_k} e(P(n',n_k))| \gg \delta. \ \ \ \ \ (7)

 

We write

\displaystyle P(n',n_k) = \sum_{i_k \leq d_k} P_{i_k}(n') n_k^i

for some polynomials {P_{i_k}: {\bf Z}^{k-1} \rightarrow {\bf R}/{\bf Z}} of degrees at most {d_1,\dots,d_{k-1}} in the variables {n_1,\dots,n_{k-1}}. For each {n'} obeying (7), we apply Corollary 5 to conclude that there exists a natural number {q_{n'} \ll \delta^{-O(1)}} such that

\displaystyle \| q_{n'} P_{i_k}(n') \|_{{\bf R}/{\bf Z}} \ll \delta^{-O(1)} / N_k^{i_k}

for {i_k=1,\dots,d_k} (the claim also holds for {i_k=0} but we discard it as being trivial). By the pigeonhole principle, there thus exists a natural number {q \ll \delta^{-O(1)}} such that

\displaystyle \| q P_{i_k}(n') \|_{{\bf R}/{\bf Z}} \ll \delta^{-O(1)} / N_k^{i_k}

for all {i_k=1,\dots,d_k} and for {\gg \delta^{O(1)} N_1 \dots N_{k-1}} choices of {n' \in Q_1 \times \dots \times Q_{k-1}}. If we write

\displaystyle P_{i_k}(n') = \sum_{i_{k-1} \leq d_{k-1}} P_{i_{k-1},i_k}(n_1,\dots,n_{k-2}) n_{k-1}^{i_{k-1}},

where {P_{i_{k-1},i_k}: {\bf Z}^{k-2} \rightarrow {\bf R}/{\bf Z}} is a polynomial of degrees at most {d_1,\dots,d_{k-2}}, then for {\gg \delta^{O(1)} N_1 \dots N_{k-2}} choices of {(n_1,\dots,n_{k-2}) \in Q_1 \times \dots \times Q_{k-2}} we then have

\displaystyle \| \sum_{i_{k-1} \leq d_{k-1}} q P_{i_{k-1},i_k}(n_1,\dots,n_{k-2}) n_{k-1}^{i_{k-1}} \|_{{\bf R}/{\bf Z}} \ll \delta^{-O(1)} / N_k^{i_k}.

Applying Lemma 6 in the {n_{k-1}} and the largeness hypotheses on the {N_i} (and also the assumption that {i_k \geq 1}) we conclude (after enlarging {q} as necessary, and pigeonholing to keep {q} independent of {n_1,\dots,n_{k-2}}) that

\displaystyle \| q P_{i_{k-1},i_k}(n_1,\dots,n_{k-2}) \|_{{\bf R}/{\bf Z}} \ll \frac{\delta^{-O(1)}}{N_{k-1}^{i_{k-1}} N_k^{i_k}}

for all {i_{k-1}=0,\dots,d_{k-1}} (note that we now include that {i_{k-1}=0} case, which is no longer trivial) and for {\gg \delta^{O(1)} N_1 \dots N_{k-2}} choices of {(n_1,\dots,n_{k-2}) \in Q_1 \times \dots \times Q_{k-2}}. Iterating this, we eventually conclude (after enlarging {q} as necessary) that

\displaystyle \| q \alpha_{i_1,\dots,i_k} \|_{{\bf R}/{\bf Z}} \ll \frac{\delta^{-O(1)}}{N_1^{i_1} \dots N_k^{i_k}} \ \ \ \ \ (8)

 

whenever {i_j \in \{0,\dots,d_j\}} for {j=1,\dots,k}, with {i_k} nonzero. Permuting the indices, and observing that the claim is trivial for {(i_1,\dots,i_k) = (0,\dots,0)}, we in fact obtain (8) for all {(i_1,\dots,i_k) \in \{0,\dots,d_1\} \times \dots \times \{0,\dots,d_k\}}, at which point the claim easily follows by taking {Q'_j := \{ qn_j: n_j \leq \delta^C N_j\}} for each {j=1,\dots,k}. \Box

An inspection of the proof of the above result (or alternatively, by combining the above result again with many applications of Lemma 6) reveals the following general form of Proposition 4, which was posed as Exercise 17 in this previous blog post, but had a slight misprint in it (it did not properly treat the possibility that some of the {N_j} could be small) and was a bit trickier to prove than anticipated (in fact, the reason for this post was that I was asked to supply a more detailed solution for this exercise):

Proposition 8 (Multidimensional Weyl exponential sum estimate, inverse form, II) Let {k \geq 1} be an natural number, and for each {j=1,\dots,k}, let {I_j \subset [0,N_j]_{\bf Z}} be a discrete interval for some {N_j \geq 1}. Let

\displaystyle P(n_1,\dots,n_k) = \sum_{i_1 \leq d_1, \dots, i_k \leq d_k} \alpha_{i_1,\dots,i_k} n_1^{i_1} \dots n_k^{i_k}

be a polynomial in {k} variables of multidegrees {d_1,\dots,d_k \geq 0} for some {\alpha_{i_1,\dots,i_k} \in {\bf R}/{\bf Z}}. If

\displaystyle \frac{1}{N_1 \dots N_k} |\sum_{n \in I_1 \times \dots \times I_k} e(P(n))| \geq \delta \ \ \ \ \ (9)

 

for some {\delta > 0}, then either

\displaystyle N_j \ll_{k,d_1,\dots,d_k} \delta^{-O_{k,d_1,\dots,d_k}(1)} \ \ \ \ \ (10)

 

for some {1 \leq j \leq d_k}, or else there is a natural number {q \ll_{k,d_1,\dots,d_k} \delta^{-O_{k,d_1,\dots,d_k}(1)}} such that

\displaystyle \| q\alpha_{i_1,\dots,i_k} \|_{{\bf R}/{\bf Z}} \ll_{k,d_1,\dots,d_k} \delta^{-O_d(1)} N_1^{-i_1} \dots N_k^{-i_k} \ \ \ \ \ (11)

 

whenever {i_j \leq d_j} for {j=1,\dots,k}.

Again, the factor of {N_1^{-i_1} \dots N_k^{-i_k}} is natural in this bound. In the {k=1} case, the option (10) may be deleted since (11) trivially holds in this case, but this simplification is no longer available for {k>1} since one needs (10) to hold for all {j} (not just one {j}) to make (11) completely trivial. Indeed, the above proposition fails for {k \geq 2} if one removes (10) completely, as can be seen for instance by inspecting the exponential sum {\sum_{n_1 \in \{0,1\}} \sum_{n_2 \in [1,N] \cap {\bf Z}} e( \alpha n_1 n_2)}, which has size comparable to {N} regardless of how irrational {\alpha} is.

In Notes 5, we saw that the Gowers uniformity norms on vector spaces {{\bf F}^n} in high characteristic were controlled by classical polynomial phases {e(\phi)}.

Now we study the analogous situation on cyclic groups {{\bf Z}/N{\bf Z}}. Here, there is an unexpected surprise: the polynomial phases (classical or otherwise) are no longer sufficient to control the Gowers norms {U^{s+1}({\bf Z}/N{\bf Z})} once {s} exceeds {1}. 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 {n \mapsto g^n x} on nilmanifolds {G/\Gamma}; 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 {n \mapsto g(n) \Gamma}, and this is the perspective we will take here.

A polynomial phase {n \mapsto e(\phi(n))} on a finite abelian group {H} is formed by starting with a polynomial {\phi: H \rightarrow {\bf R}/{\bf Z}} to the unit circle, and then composing it with the exponential function {e: {\bf R}/{\bf Z} \rightarrow {\bf C}}. To create a nilsequence {n \mapsto F(g(n) \Gamma)}, we generalise this construction by starting with a polynomial {g \Gamma: H \rightarrow G/\Gamma} into a nilmanifold {G/\Gamma}, and then composing this with a Lipschitz function {F: G/\Gamma \rightarrow {\bf C}}. (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 {n \mapsto e( \lfloor \alpha n \rfloor \beta n )}. (The “almost” here is because the relevant functions {F: G/\Gamma \rightarrow {\bf C}} 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.

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In the previous lectures, we have focused mostly on the equidistribution or linear patterns on a subset of the integers {{\bf Z}}, and in particular on intervals {[N]}. The integers are of course a very important domain to study in additive combinatorics; but there are also other fundamental model examples of domains to study. One of these is that of a vector space {V} over a finite field {{\bf F} = {\bf F}_p} of prime order. Such domains are of interest in computer science (particularly when {p=2}) and also in number theory; but they also serve as an important simplified “dyadic model” for the integers. See this survey article of Green for further discussion of this point.

The additive combinatorics of the integers {{\bf Z}}, and of vector spaces {V} over finite fields, are analogous, but not quite identical. For instance, the analogue of an arithmetic progression in {{\bf Z}} is a subspace of {V}. In many cases, the finite field theory is a little bit simpler than the integer theory; for instance, subspaces are closed under addition, whereas arithmetic progressions are only “almost” closed under addition in various senses. (For instance, {[N]} is closed under addition approximately half of the time.) However, there are some ways in which the integers are better behaved. For instance, because the integers can be generated by a single generator, a homomorphism from {{\bf Z}} to some other group {G} can be described by a single group element {g}: {n \mapsto g^n}. However, to specify a homomorphism from a vector space {V} to {G} one would need to specify one group element for each dimension of {V}. Thus we see that there is a tradeoff when passing from {{\bf Z}} (or {[N]}) to a vector space model; one gains a bounded torsion property, at the expense of conceding the bounded generation property. (Of course, if one wants to deal with arbitrarily large domains, one has to concede one or the other; the only additive groups that have both bounded torsion and boundedly many generators, are bounded.)

The starting point for this course (Notes 1) was the study of equidistribution of polynomials {P: {\bf Z} \rightarrow {\bf R}/{\bf Z}} from the integers to the unit circle. We now turn to the parallel theory of equidistribution of polynomials {P: V \rightarrow {\bf R}/{\bf Z}} from vector spaces over finite fields to the unit circle. Actually, for simplicity we will mostly focus on the classical case, when the polynomials in fact take values in the {p^{th}} roots of unity (where {p} is the characteristic of the field {{\bf F} = {\bf F}_p}). As it turns out, the non-classical case is also of importance (particularly in low characteristic), but the theory is more difficult; see these notes for some further discussion.

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(Linear) Fourier analysis can be viewed as a tool to study an arbitrary function {f} on (say) the integers {{\bf Z}}, by looking at how such a function correlates with linear phases such as {n \mapsto e(\xi n)}, where {e(x) := e^{2\pi i x}} is the fundamental character, and {\xi \in {\bf R}} is a frequency. These correlations control a number of expressions relating to {f}, such as the expected behaviour of {f} on arithmetic progressions {n, n+r, n+2r} of length three.

In this course we will be studying higher-order correlations, such as the correlation of {f} with quadratic phases such as {n \mapsto e(\xi n^2)}, as these will control the expected behaviour of {f} on more complex patterns, such as arithmetic progressions {n, n+r, n+2r, n+3r} of length four. In order to do this, we must first understand the behaviour of exponential sums such as

\displaystyle  \sum_{n=1}^N e( \alpha n^2 ).

Such sums are closely related to the distribution of expressions such as {\alpha n^2 \hbox{ mod } 1} in the unit circle {{\bf T} := {\bf R}/{\bf Z}}, as {n} varies from {1} to {N}. More generally, one is interested in the distribution of polynomials {P: {\bf Z}^d \rightarrow {\bf T}} of one or more variables taking values in a torus {{\bf T}}; for instance, one might be interested in the distribution of the quadruplet {(\alpha n^2, \alpha (n+r)^2, \alpha(n+2r)^2, \alpha(n+3r)^2)} as {n,r} both vary from {1} to {N}. Roughly speaking, once we understand these types of distributions, then the general machinery of quadratic Fourier analysis will then allow us to understand the distribution of the quadruplet {(f(n), f(n+r), f(n+2r), f(n+3r))} for more general classes of functions {f}; this can lead for instance to an understanding of the distribution of arithmetic progressions of length {4} in the primes, if {f} is somehow related to the primes.

More generally, to find arithmetic progressions such as {n,n+r,n+2r,n+3r} in a set {A}, it would suffice to understand the equidistribution of the quadruplet {(1_A(n), 1_A(n+r), 1_A(n+2r), 1_A(n+3r))} in {\{0,1\}^4} as {n} and {r} vary. This is the starting point for the fundamental connection between combinatorics (and more specifically, the task of finding patterns inside sets) and dynamics (and more specifically, the theory of equidistribution and recurrence in measure-preserving dynamical systems, which is a subfield of ergodic theory). This connection was explored in one of my previous classes; it will also be important in this course (particularly as a source of motivation), but the primary focus will be on finitary, and Fourier-based, methods.

The theory of equidistribution of polynomial orbits was developed in the linear case by Dirichlet and Kronecker, and in the polynomial case by Weyl. There are two regimes of interest; the (qualitative) asymptotic regime in which the scale parameter {N} is sent to infinity, and the (quantitative) single-scale regime in which {N} is kept fixed (but large). Traditionally, it is the asymptotic regime which is studied, which connects the subject to other asymptotic fields of mathematics, such as dynamical systems and ergodic theory. However, for many applications (such as the study of the primes), it is the single-scale regime which is of greater importance. The two regimes are not directly equivalent, but are closely related: the single-scale theory can be usually used to derive analogous results in the asymptotic regime, and conversely the arguments in the asymptotic regime can serve as a simplified model to show the way to proceed in the single-scale regime. The analogy between the two can be made tighter by introducing the (qualitative) ultralimit regime, which is formally equivalent to the single-scale regime (except for the fact that explicitly quantitative bounds are abandoned in the ultralimit), but resembles the asymptotic regime quite closely.

We will view the equidistribution theory of polynomial orbits as a special case of Ratner’s theorem, which we will study in more generality later in this course.

For the finitary portion of the course, we will be using asymptotic notation: {X \ll Y}, {Y \gg X}, or {X = O(Y)} denotes the bound {|X| \leq CY} for some absolute constant {C}, and if we need {C} to depend on additional parameters then we will indicate this by subscripts, e.g. {X \ll_d Y} means that {|X| \leq C_d Y} for some {C_d} depending only on {d}. In the ultralimit theory we will use an analogue of asymptotic notation, which we will review later in these notes.

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Today, Prof. Margulis continued his lecture series, focusing on two specific examples of homogeneous dynamics applications to number theory, namely counting lattice points on algebraic varieties, and quantitative versions of the Oppenheim conjecture.  (Due to lack of time, the third application mentioned in the previous lecture, namely metric theory of Diophantine approximation, was not covered.)

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The final distinguished lecture series for the academic year here at UCLA is being given this week by Gregory Margulis, who is giving three lectures on “homogeneous dynamics and number theory”.  In his first lecture, Prof. Margulis surveyed some classical problems in number theory that turn out, rather surprisingly, to have more or less equivalent counterparts in homogeneous dynamics – the theory of dynamical systems on homogeneous spaces G/\Gamma.

As usual, any errors in this post are due to my transcription of the talk.

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This week I was in Columbus, Ohio, attending a conference on equidistribution on manifolds. I talked about my recent paper with Ben Green on the quantitative behaviour of polynomial sequences in nilmanifolds, which I have blogged about previously. During my talk (and inspired by the immediately preceding talk of Vitaly Bergelson), I stated explicitly for the first time a generalisation of the van der Corput trick which morally underlies our paper, though it is somewhat buried there as we specialised it to our application at hand (and also had to deal with various quantitative issues that made the presentation more complicated). After the talk, several people asked me for a more precise statement of this trick, so I am presenting it here, and as an application reproving an old theorem of Leon Green that gives a necessary and sufficient condition as to whether a linear sequence (g^n x)_{n=1}^\infty on a nilmanifold G/\Gamma is equidistributed, which generalises the famous theorem of Weyl on equidistribution of polynomials.

UPDATE, Feb 2013: It has been pointed out to me by Pavel Zorin that this argument does not fully recover the theorem of Leon Green; to cover all cases, one needs the more complicated van der Corput argument in our paper.

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Ben Green and I have just uploaded our joint paper, “The distribution of polynomials over finite fields, with applications to the Gowers norms“, to the arXiv, and submitted to Contributions to Discrete Mathematics. This paper, which we first announced at the recent FOCS meeting, and then gave an update on two weeks ago on this blog, is now in final form. It is being made available simultaneously with a closely related paper of Lovett, Meshulam, and Samorodnitsky.

In the previous post on this topic, I focused on the negative results in the paper, and in particular the fact that the inverse conjecture for the Gowers norm fails for certain degrees in low characteristic. Today, I’d like to focus instead on the positive results, which assert that for polynomials in many variables over finite fields whose degree is less than the characteristic of the field, one has a satisfactory theory for the distribution of these polynomials. Very roughly speaking, the main technical results are:

  • A regularity lemma: Any polynomial can be expressed as a combination of a bounded number of other polynomials which are regular, in the sense that no non-trivial linear combination of these polynomials can be expressed efficiently in terms of lower degree polynomials.
  • A counting lemma: A regular collection of polynomials behaves as if the polynomials were selected randomly. In particular, the polynomials are jointly equidistributed.

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Ben Green and I have just uploaded our paper “The quantitative behaviour of polynomial orbits on nilmanifolds” to the arXiv (and shortly to be submitted to a journal, once a companion paper is finished). This paper grew out of our efforts to prove the Möbius and Nilsequences conjecture MN(s) from our earlier paper, which has applications to counting various linear patterns in primes (Dickson’s conjecture). These efforts were successful – as the companion paper will reveal – but it turned out that in order to establish this number-theoretic conjecture, we had to first establish a purely dynamical quantitative result about polynomial sequences in nilmanifolds, very much in the spirit of the celebrated theorems of Marina Ratner on unipotent flows; I plan to discuss her theorems in more detail in a followup post to this one.In this post I will not discuss the number-theoretic applications or the connections with Ratner’s theorem, and instead describe our result from a slightly different viewpoint, starting from some very simple examples and gradually moving to the general situation considered in our paper.

To begin with, consider a infinite linear sequence (n \alpha + \beta)_{n \in {\Bbb N}} in the unit circle {\Bbb R}/{\Bbb Z}, where \alpha, \beta \in {\Bbb R}/{\Bbb Z}. (One can think of this sequence as the orbit of \beta under the action of the shift operator T: x \mapsto x +\alpha on the unit circle.) This sequence can do one of two things:

  1. If \alpha is rational, then the sequence (n \alpha + \beta)_{n \in {\Bbb N}} is periodic and thus only takes on finitely many values.
  2. If \alpha is irrational, then the sequence (n \alpha + \beta)_{n \in {\Bbb N}} is dense in {\Bbb R}/{\Bbb Z}. In fact, it is not just dense, it is equidistributed, or equivalently that

    \displaystyle\lim_{N \to \infty} \frac{1}{N} \sum_{n=1}^N F( n \alpha + \beta ) = \int_{{\Bbb R}/{\Bbb Z}} F

    for all continuous functions F: {\Bbb R}/{\Bbb Z} \to {\Bbb C}. This statement is known as the equidistribution theorem.

We thus see that infinite linear sequences exhibit a sharp dichotomy in behaviour between periodicity and equidistribution; intermediate scenarios, such as concentration on a fractal set (such as a Cantor set), do not occur with linear sequences. This dichotomy between structure and randomness is in stark contrast to exponential sequences such as ( 2^n \alpha)_{n \in {\Bbb N}}, which can exhibit an extremely wide spectrum of behaviours. For instance, the question of whether (10^n \pi)_{n \in {\Bbb N}} is equidistributed mod 1 is an old unsolved problem, equivalent to asking whether \pi is normal base 10.

Intermediate between linear sequences and exponential sequences are polynomial sequences (P(n))_{n \in {\Bbb N}}, where P is a polynomial with coefficients in {\Bbb R}/{\Bbb Z}. A famous theorem of Weyl asserts that infinite polynomial sequences enjoy the same dichotomy as their linear counterparts, namely that they are either periodic (which occurs when all non-constant coefficients are rational) or equidistributed (which occurs when at least one non-constant coefficient is irrational). Thus for instance the fractional parts \{ \sqrt{2}n^2\} of \sqrt{2} n^2 are equidistributed modulo 1. This theorem is proven by Fourier analysis combined with non-trivial bounds on Weyl sums.

For our applications, we are interested in strengthening these results in two directions. Firstly, we wish to generalise from polynomial sequences in the circle {\Bbb R}/{\Bbb Z} to polynomial sequences (g(n)\Gamma)_{n \in {\Bbb N}} in other homogeneous spaces, in particular nilmanifolds. Secondly, we need quantitative equidistribution results for finite orbits (g(n)\Gamma)_{1 \leq n \leq N} rather than qualitative equidistribution for infinite orbits (g(n)\Gamma)_{n \in {\Bbb N}}.

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