Asgar Jamneshan, Or Shalom, and myself have just uploaded to the arXiv our preprint “The structure of arbitrary Conze–Lesigne systems“. As the title suggests, this paper is devoted to the structural classification of Conze-Lesigne systems, which are a type of measure-preserving system that are “quadratic” or of “complexity two” in a certain technical sense, and are of importance in the theory of multiple recurrence. There are multiple ways to define such systems; here is one. Take a countable abelian group 
is a Folner sequence for and 
is the complex conjugation map. One can show that this limit exists and is independent of the choice of Folner sequence, and that the 
seminorm is indeed a seminorm. A Conze-Lesigne system is an ergodic measure-preserving system in which the 
seminorm is in fact a norm, thus 
whenever is non-zero. Informally, this means that when one considers a generic parallelepiped in a Conze–Lesigne system 
, the location of any vertex of that parallelepiped is more or less determined by the location of the other seven vertices. These are the important systems to understand in order to study “complexity two” patterns, such as arithmetic progressions of length four. While not all systems are Conze-Lesigne systems, it turns out that they always have a maximal factor 
that is a Conze-Lesigne system, known as the Conze-Lesigne factor or the second Host-Kra-Ziegler factor of the system, and this factor controls all the complexity two recurrence properties of the system.
The analogous theory in complexity one is well understood. Here, one replaces the 
is a norm are called Kronecker systems. These systems are completely classified: a system is Kronecker if and only if it arises from a compact abelian group 
equipped with Haar probability measure and a translation action 
for some homomorphism 
with dense image. Such systems can then be analyzed quite efficiently using the Fourier transform, and this can then be used to satisfactory analyze “complexity one” patterns, such as length three progressions, in arbitrary systems (or, when translated back to combinatorial settings, in arbitrary dense sets of abelian groups).
We return now to the complexity two setting. The most famous examples of Conze-Lesigne systems are (order two) nilsystems, in which the space 
two real numbers with 
linearly independent over 
, is a Conze-Lesigne system. As the base case of a well known result of Host and Kra, it is shown in fact that all Conze-Lesigne -systems are inverse limits of nilsystems (previous results in this direction were obtained by Conze-Lesigne, Furstenberg-Weiss, and others). Similar results are known for -systems when is finitely generated, thanks to the thesis work of Griesmer (with further proofs by Gutman-Lian and Candela-Szegedy). However, this is not the case once is not finitely generated; as a recent example of Shalom shows, Conze-Lesigne systems need not be the inverse limit of nilsystems in this case.
Our main result is that even in the infinitely generated case, Conze-Lesigne systems are still inverse limits of a slight generalisation of the nilsystem concept, in which
Theorem 1 (Classification of Conze-Lesigne systems) Let), with
In a forthcoming companion paper to this one, Asgar Jamneshan and I will use this theorem to derive an inverse theorem for the Gowers norm 
We sketch some of the main ideas used to prove the theorem. The existing machinery developed by Conze-Lesigne, Furstenberg-Weiss, Host-Kra, and others allows one to describe an arbitrary Conze-Lesigne system as a group extension 
and almost all 
, and some measurable function 
; roughly speaking this asserts that 
is “linear up to coboundaries”. For technical reasons it is also convenient to reduce to the case where is separable. The problem is that the equation (2) is unwieldy to work with. In the model case when the target group 
is a circle 
, one can use some Fourier analysis to convert (2) into the more tractable Conze-Lesigne equation 
, all 
, and almost all , where for each 
, 
is a measurable function, and 
is a homomorphism. (For technical reasons it is often also convenient to enforce that 
depend in a measurable fashion on ; this can always be achieved, at least when the Conze-Lesigne system is separable, but actually verifying that this is possible actually requires a certain amount of effort, which we devote an appendix to in our paper.) It is not difficult to see that (3) implies (2) for any group (as long as one has the measurability in mentioned previously), but the converse turns out to fail for some groups , such as solenoid groups (e.g., inverse limits of 
as 
), as was essentially shown by Rudolph. However, in our paper we were able to find a separate argument that also derived the Conze-Lesigne equation in the case of a cyclic group 
. Putting together the 
and cases, one can then derive the Conze-Lesigne equation for arbitrary compact abelian Lie groups (as such groups are isomorphic to direct products of finitely many tori and cyclic groups). As has been known for some time (see e.g., this paper of Host and Kra), once one has a Conze-Lesigne equation, one can more or less describe the system as a translational system , where the Host-Kra group is the set of all pairs 
that solve an equation of the form (3) (with these pairs acting on 
by the law 
), and is the stabiliser of a point in this system. This then establishes the theorem in the case when is a Lie group, and the general case basically comes from the fact (from Fourier analysis or the Peter-Weyl theorem) that an arbitrary compact abelian group is an inverse limit of Lie groups. (There is a technical issue here in that one has to check that the space of translational system factors of form a directed set in order to have a genuine inverse limit, but this can be dealt with by modifications of the tools mentioned here.)
There is an additional technical issue worth pointing out here (which unfortunately was glossed over in some previous work in the area). Because the cocycle equation (1) and the Conze-Lesigne equation (3) are only valid almost everywhere instead of everywhere, the action of
Now we discuss how to establish the Conze-Lesigne equation (3) in the cyclic group case 
together with their intersection 
and 
one might use a Venn diagram such as
(or measure 
) of the set 
corresponding to such a region. For instance, the above Venn diagram can be used to intuitively justify the inclusion-exclusion formula 
, the image 
of that function, and the image 
of some subset 
of 
; one can similarly illustrate injectivity of 
by giving 
:
, such as the reals) or abelian groups, rather than sets. The categories of (
accurate or 
accurate, that there is a 
chance of dying from some disease, and so forth. However, for those without extensive mathematical training, it can sometimes be difficult to assess whether a given percentage amounts to a “good” or “bad” outcome, because this depends very much on the context of how the percentage is used. For instance:
to 
might be considered close, but 
to 
would probably be viewed as a convincing mandate, and 
to 
would likely be viewed as a landslide. 
or 
likely to be non-fatal (i.e., a 
or 
chance of death) would not be conducted lightly. 
chance of rain during a vacation trip might be sufficient cause to pack an umbrella, even though it is more likely than not that rain would not occur. On the other hand, if the prediction was for an 
(15 parts per billion). At the opposite extreme, recycling contamination rates as high as 
are in the probability of successfully avoiding the negative outcome, thus
success = two nines of safety 
success = three nines of safety
of the “safe” outcome and probability 
of the “unsafe” outcome will have 
is the logarithm to base ten), or equivalently 
. In an ideal world, one would have infinitely many nines of safety against any risk, but in practice there are no 
(approx.) 
, or about 
, or about 
nines of safety.
per 
, providing only 
nines of safety. Of course, the safety will decrease with each additional round: one has only 
nines of safety after two rounds, 
nines after three rounds, and so forth. (See also Proposition 
on a 
-sided die is a risk that carries about 
statistically significant result. (However, one has to be careful when reversing the conditional; a 
nines of safety against that opponent being dealt a royal flush, 
against a straight flush or higher, 
against a full house or higher, 
against a flush or higher, 
against a straight or higher, 
against three-of-a-kind or higher, 
against two pairs or higher, and just 
nines of safety against a single guess. (For the reduction in safety caused by multiple guesses, see Proposition 
. 
trials is 
, which is a 
. The claim then follows from the standard properties of that distribution. 
times, then (assuming 
nines of safety. Conversely: if the repeated activity has 
nines of safety, the individual activity will have approximately 
nines of safety. 
, hence safe with probability 
if repeated independently 
; but in this case there is perfect correlation, and in fact the number of nines of safety remains steady at 
nines, the weekly amount to be about 
nines, and the amount of safety over five years to drop to about 
nines.
people) 
people) 
people 
people 
people 
people 
nines of safety against this risk, your family of five might only have about 
nines of safety, and so forth. By the time one gets to a group of 
is not possible, but one should interpret a prediction of a negative number of nines as an assertion that failure is very likely to happen. Also, in practice the reduction in safety is less than this rule predicts, due to correlations such as risk factors that are common to the group being considered that are incompatible with the assumption of independence.)
.
nines of safety (that is to say, the probability that the protection is effective is 
). Then applying that precaution increases the number of nines in the activity from 
. 
. The claim now follows from Definition 
effectiveness against symptomatic COVID illness, providing about 
nines of safety against that risk, and over 
. Then the action removes 
nines of safety (if 
) or adds 
nines of safety (if 
) to the original activity. 
. The claim now follows from Definition 
, adding about 
additional nines of safety. 
, and let 
. Then the “one-sided risk” of 
by at least 
(i.e., 
) carries 
) carries 
is the 
of deviations from the mean 
of a Dirichlet 
-function 
corresponding to a primitive quadratic character 
of some conductor 
, which is close to 
(which we will call the quality) of the Siegel zero. The significance of these zeroes is that they force the Möbius function 
and the Liouville function 
in which 
, then very few primes near 
and 
, where the sum is over exceptional primes in the indicated range 
; this bound is non-trivial for 
as large as 
. (See Section 1 of 
.
be a fixed natural number. Suppose one has a Siegel zero 
, where 
is the singular series 
be distinct fixed natural numbers. Suppose one has a Siegel zero 
, where 
be distinct fixed natural numbers with 
. Suppose one has a Siegel zero 
, which is a completely multiplicative function that agrees with 
, which is the Dirichlet convolution 
multiplied by a 
to essentially restrict that convolution to almost primes. 
which has the structure of a “Type I sum”, and which agrees with 
which has the structure of a “Type I sum”. 
, and the types of correlation one is faced with at the start of step (iv) are a more complicated version of the divisor correlation sum 
this sum can be easily controlled by the 
one needs the fact that 
; in fact Kloosterman sum bounds give a level of distribution of 
, a folklore fact that seems to have first been observed by Linnik and Selberg. We use a (notationally more complicated) version of this argument to treat the sums arising in (iv) for 
there are no known techniques to unconditionally obtain asymptotics, even for the model sum 
which is nowhere analytic, but is such that the Taylor series 
converges for every 
? (Of course, this series would not then converge to 
for each 
.) I have a vague feeling that perhaps the Baire category theorem should be able to resolve this question, but it seems to require a bit of effort. (Update: answered by Alexander Shaposhnikov in comments.) 
to infinite order in the following sense: for every polynomial 
, there exists 
for all 
? Such a function would be rather pathological, perhaps resembling a space-filling curve. (Update: solved for smooth 
that diverges everywhere (except at 
), but which becomes pointwise convergent after dividing each of the monomials 
into pieces 
for some 
summing absolutely to 
, and then rearranging, i.e., there is some rearrangement 
of 
that is pointwise convergent for every 
of 
in 
of the tiles 
to the (translational) tiling equation 
of 
denotes the set of sums 
if the sums 
are all disjoint (and is undefined otherwise), and 
denotes disjoint union. (One can also write the tiling equation in the language of convolutions as 
.)
, one can ask whether the solvability of the tiling equation 
by appropriate constructions in ZFC). If this is the case we say that the tiling equation 
in some specified class (and encoded somehow as binary strings), one can ask whether the solvability of the tiling equation 
, whereas the notion of algorithmic decidability pertains instead to classes of data, and is only interesting when this class is infinite. Indeed, any tiling problem with a finite class of data is trivially decidable because one could simply code a Turing machine that is basically a lookup table that returns the correct answer for each choice of data in the class. (This is akin to how a student with a good memory could pass any exam if the questions are drawn from a finite list, merely by memorising an answer key for that list of questions.)
of sets that we believe satisfies the axioms of ZFC, there are also other “nonstandard” universes 
that also obey the axioms of ZFC. If the solvability of a tiling equation 
where all of the 
, and using rotations as well as translations) come from the 
(or more generally 
with 
a finite group), a simple pigeonholing argument shows that no tiling equations are aperiodic, and hence all tiling equations are decidable. However the situation changes in two dimensions. In 1966, Berger (a student of Wang) 
that are aperiodic, or even logically undecidable; in fact he showed that the tiling problem in this case (with arbitrary choices of data 
) was algorithmically undecidable. (Strictly speaking, Berger established this for a variant of the tiling problem known as the domino problem, but later 
required for Berger’s construction was quite large: his construction of an aperiodic tiling required 
tiles, and his construction of a logically undecidable tiling required an even larger (and not explicitly specified) collection of tiles. Subsequent work by many authors did reduce the number of tiles required; in the 
setting, the current world record for the fewest number of tiles in an aperiodic tiling is 
(due 
(due 
is a finite subset of 
is not aperiodic. 
, and more recently 
), but remains open in higher dimensions. By the preceding discussion, the conjecture implies that every tiling equation with a single tile is logically decidable, and the problem of whether a given periodic set can be tiled by a single tile is algorithmically decidable.
, at least if one is permitted to enlarge the group 
for some finite abelian 
of 
such that the tiling equation 
is logically undecidable (and hence also aperiodic). 
, a periodic subset 
, and finite sets 
is logically undecidable (and hence also aperiodic). 
such that the nonabelian tiling equation 
is logically undecidable (and hence also aperiodic).
is algorithmically undecidable. Similarly for (ii), (iii).
theory of tiling by a single (abelian) tile, and the 
theory of tiling with multiple tiles (or one non-abelian tile). (The positive results on the periodic tiling conjecture certainly rely heavily on the fact that there is only one tile, in particular there is a “dilation lemma” that is only available in this setting that is of key importance in the two dimensional theory.) It would be nice to eliminate the group 
in (ii)), but I think this would require a fairly significant modification of our methods.
of a finitely generated group 
and 
, as well as periodic sets 
of 
, we can “stack” these equations into a single tiling equation in the larger group 
, and specifically to the equation 
taking values in a finite group 
is a coset of some subgroup 
of 
for all 
, if we desire. If 
, and 
, then the additional equation 
for all 
where each of the four sides is colored with a color 
(corresponding to the four cardinal directions North, South, East, and West) from some finite set 
of colors. The domino problem is then to tile the plane with copies of these tiles in such a way that adjacent sides match. In terms of equations, one is seeking to find functions 
obeying the pointwise constraint 
is the set of colors associated to the set of Wang tiles being used, and the matching constraints 
. As it turns out, the pointwise constraint 
, but in order to encode the matching constraints 
to this equation that agree on the left in the sense that 
, define the “swap” 
of 
and 
to be the set 
, 
at each position 
, and 
(say), which is a toy version of the matching constraint 
attached to it, known as the 
of global sections may vanish. But as one raises the canonical line bundle 
, the number of global sections tends to increase; in particular, the dimension 
of global sections (known as the 
for some non-negative number 
, which is called the volume of the variety 
(roughly speaking, the number of common zeroes of 
given 
given by 
; the method also gives comparable bounds for some other related algebraic geometry statistics, such as the largest 
is not a birational embedding into projective space.
and resolving the singularities. These varieties are relatively tractable to work with, as one can use standard algebraic geometry tools (such as the 
. The problem then reduces to optimizing the resulting volume given the constraints needed for the above-mentioned sufficient conditions to hold. After working with a particular choice of weights (which consist of products of mostly consecutive primes, with each product occuring with suitable multiplicities 
), the problem eventually boils down to trying to minimize the total multiplicity 
, subject to certain congruence conditions and other bounds on the 
. Using crude bounds on the 
, but by taking advantage of the ability to “dilate” the congruence conditions and optimizing over all dilations, we are able to improve the 
.
![{g: {\bf R} \rightarrow (-1/2,1/2]}](https://s0.wp.com/latex.php?latex=%7Bg%3A+%7B%5Cbf+R%7D+%5Crightarrow+%28-1%2F2%2C1%2F2%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
, with 
defined as the unique real number in ![{(-1/2,1/2]}](https://s0.wp.com/latex.php?latex=%7B%28-1%2F2%2C1%2F2%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
that is equal to 
is bounded above by 
could attain the value of 
would vanish. For the algebraic geometry application discussed above one is then led to the following question: for a given choice of 
on the quantity 
that holds for all probability measures 
such that 
of an integer, so we have 
. In fact we were able to compute this quantity precisely:
.
for some natural number 
, then 
. 
for some natural number 
.
as 
such that one had the pointwise bound 
. The theory of 
to the numbers 
and the remaining probability of 
to 
, while the optimal weighted inequality 
for the inequality 
with 
odd, with probability proportional to 
, and the lower bound verification eventually follows from a classical identity 
for 
. For finitely supported functions 
, one can define the (non-normalised) Gowers norm 
by the formula 
![{[N] = \{1,\dots,N\}}](https://s0.wp.com/latex.php?latex=%7B%5BN%5D+%3D+%5C%7B1%2C%5Cdots%2CN%5C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
and any function ![{f: [N] \rightarrow {\bf C}}](https://s0.wp.com/latex.php?latex=%7Bf%3A+%5BN%5D+%5Crightarrow+%7B%5Cbf+C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
we can then define the (normalised) Gowers norm ![\displaystyle \|f\|_{U^k([N])} := \| f 1_{[N]} \|_{\tilde U^k({\bf Z})} / \|1_{[N]} \|_{\tilde U^k({\bf Z})}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5C%7Cf%5C%7C_%7BU%5Ek%28%5BN%5D%29%7D+%3A%3D+%5C%7C+f+1_%7B%5BN%5D%7D+%5C%7C_%7B%5Ctilde+U%5Ek%28%7B%5Cbf+Z%7D%29%7D+%2F+%5C%7C1_%7B%5BN%5D%7D+%5C%7C_%7B%5Ctilde+U%5Ek%28%7B%5Cbf+Z%7D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002)
![{f 1_{[N]}: {\bf Z} \rightarrow {\bf C}}](https://s0.wp.com/latex.php?latex=%7Bf+1_%7B%5BN%5D%7D%3A+%7B%5Cbf+Z%7D+%5Crightarrow+%7B%5Cbf+C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
is the extension of ![\displaystyle \|f\|_{U^1([N])} = |\mathop{\bf E}_{n \in [N]} f(n)|](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5C%7Cf%5C%7C_%7BU%5E1%28%5BN%5D%29%7D+%3D+%7C%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+f%28n%29%7C&bg=ffffff&fg=000000&s=0&c=20201002)
![{\| \|_{U^1([N])}}](https://s0.wp.com/latex.php?latex=%7B%5C%7C+%5C%7C_%7BU%5E1%28%5BN%5D%29%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
a seminorm rather than a norm), and one can calculate ![\displaystyle \|f\|_{U^2([N])} \asymp (N \int_0^1 |\mathop{\bf E}_{n \in [N]} f(n) e(-\alpha n)|^4\ d\alpha)^{1/4} \ \ \ \ \ (1)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5C%7Cf%5C%7C_%7BU%5E2%28%5BN%5D%29%7D+%5Casymp+%28N+%5Cint_0%5E1+%7C%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+f%28n%29+e%28-%5Calpha+n%29%7C%5E4%5C+d%5Calpha%29%5E%7B1%2F4%7D+%5C+%5C+%5C+%5C+%5C+%281%29&bg=ffffff&fg=000000&s=0&c=20201002)
, and we use the averaging notation 
.
and functions ![{f_1,\dots,f_m: [N] \rightarrow {\bf C}}](https://s0.wp.com/latex.php?latex=%7Bf_1%2C%5Cdots%2Cf_m%3A+%5BN%5D+%5Crightarrow+%7B%5Cbf+C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
, we define the multilinear form ![\displaystyle \Lambda^{P_1,\dots,P_m}(f_1,\dots,f_m) := \sum_{n \in {\bf Z}^d} \prod_{j=1}^m f_j 1_{[N]}(P_j(n)) / \sum_{n \in {\bf Z}^d} \prod_{j=1}^m 1_{[N]}(P_j(n))](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5CLambda%5E%7BP_1%2C%5Cdots%2CP_m%7D%28f_1%2C%5Cdots%2Cf_m%29+%3A%3D+%5Csum_%7Bn+%5Cin+%7B%5Cbf+Z%7D%5Ed%7D+%5Cprod_%7Bj%3D1%7D%5Em+f_j+1_%7B%5BN%5D%7D%28P_j%28n%29%29+%2F+%5Csum_%7Bn+%5Cin+%7B%5Cbf+Z%7D%5Ed%7D+%5Cprod_%7Bj%3D1%7D%5Em+1_%7B%5BN%5D%7D%28P_j%28n%29%29&bg=ffffff&fg=000000&s=0&c=20201002)
![\displaystyle \Lambda^{\mathrm{n}}(f) = \mathop{\bf E}_{n \in [N]} f(n)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5CLambda%5E%7B%5Cmathrm%7Bn%7D%7D%28f%29+%3D+%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+f%28n%29&bg=ffffff&fg=000000&s=0&c=20201002)
![\displaystyle \Lambda^{\mathrm{n}, \mathrm{n}+\mathrm{r}}(f,g) = (\mathop{\bf E}_{n \in [N]} f(n)) (\mathop{\bf E}_{n \in [N]} g(n))](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5CLambda%5E%7B%5Cmathrm%7Bn%7D%2C+%5Cmathrm%7Bn%7D%2B%5Cmathrm%7Br%7D%7D%28f%2Cg%29+%3D+%28%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+f%28n%29%29+%28%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+g%28n%29%29&bg=ffffff&fg=000000&s=0&c=20201002)
![\displaystyle \Lambda^{\mathrm{n}, \mathrm{n}+\mathrm{r}, \mathrm{n}+2\mathrm{r}}(f,g,h) \asymp \mathop{\bf E}_{n \in [N]} \mathop{\bf E}_{r \in [-N,N]} f(n) g(n+r) h(n+2r)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5CLambda%5E%7B%5Cmathrm%7Bn%7D%2C+%5Cmathrm%7Bn%7D%2B%5Cmathrm%7Br%7D%2C+%5Cmathrm%7Bn%7D%2B2%5Cmathrm%7Br%7D%7D%28f%2Cg%2Ch%29+%5Casymp+%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+%5Cmathop%7B%5Cbf+E%7D_%7Br+%5Cin+%5B-N%2CN%5D%7D+f%28n%29+g%28n%2Br%29+h%28n%2B2r%29&bg=ffffff&fg=000000&s=0&c=20201002)
![\displaystyle \Lambda^{\mathrm{n}, \mathrm{n}+\mathrm{r}, \mathrm{n}+\mathrm{r}^2}(f,g,h) \asymp \mathop{\bf E}_{n \in [N]} \mathop{\bf E}_{r \in [-N^{1/2},N^{1/2}]} f(n) g(n+r) h(n+r^2)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5CLambda%5E%7B%5Cmathrm%7Bn%7D%2C+%5Cmathrm%7Bn%7D%2B%5Cmathrm%7Br%7D%2C+%5Cmathrm%7Bn%7D%2B%5Cmathrm%7Br%7D%5E2%7D%28f%2Cg%2Ch%29+%5Casymp+%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+%5Cmathop%7B%5Cbf+E%7D_%7Br+%5Cin+%5B-N%5E%7B1%2F2%7D%2CN%5E%7B1%2F2%7D%5D%7D+f%28n%29+g%28n%2Br%29+h%28n%2Br%5E2%29&bg=ffffff&fg=000000&s=0&c=20201002)
as formal (indeterminate) variables, and ![{f,g,h: [N] \rightarrow {\bf C}}](https://s0.wp.com/latex.php?latex=%7Bf%2Cg%2Ch%3A+%5BN%5D+%5Crightarrow+%7B%5Cbf+C%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
are understood to be extended by zero to all of 
is closely related to the number of length three arithmetic progressions contained in 
is controlled by the ![{U^k[N]}](https://s0.wp.com/latex.php?latex=%7BU%5Ek%5BN%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
norm if the form is small whenever 
small in 
to be the least ![{U^{s+1}[N]}](https://s0.wp.com/latex.php?latex=%7BU%5E%7Bs%2B1%7D%5BN%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
norm. For instance,
and 
have true complexity 
has true complexity 
has true complexity 
(which among other things could be used to count twin primes) has infinite true complexity (which is quite unfortunate for applications).
; Ben Green and I thought we had resolved this conjecture 
(semi-)norm is so weak that it barely controls any averages at all. For instance the average ![\displaystyle \Lambda^{2\mathrm{n}}(f) = \mathop{\bf E}_{n \in [N], \hbox{ even}} f(n)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5CLambda%5E%7B2%5Cmathrm%7Bn%7D%7D%28f%29+%3D+%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%2C+%5Chbox%7B+even%7D%7D+f%28n%29&bg=ffffff&fg=000000&s=0&c=20201002)
![{U^1[N]}](https://s0.wp.com/latex.php?latex=%7BU%5E1%5BN%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
semi-norm: it is perfectly possible for a ![{U^1([N])}](https://s0.wp.com/latex.php?latex=%7BU%5E1%28%5BN%5D%29%7D&bg=ffffff&fg=000000&s=0&c=20201002)
norm but have large value of 
(consider for instance the parity function 
).
(or “profinite ![\displaystyle \| f\|_{U^{1^+}[N]} := \frac{1}{N} \sup_P |\sum_{n \in P} f(n)| = \sup_P | \mathop{\bf E}_{n \in [N]} f 1_P(n)|](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5C%7C+f%5C%7C_%7BU%5E%7B1%5E%2B%7D%5BN%5D%7D+%3A%3D+%5Cfrac%7B1%7D%7BN%7D+%5Csup_P+%7C%5Csum_%7Bn+%5Cin+P%7D+f%28n%29%7C+%3D+%5Csup_P+%7C+%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+f+1_P%28n%29%7C&bg=ffffff&fg=000000&s=0&c=20201002)
![{[N]}](https://s0.wp.com/latex.php?latex=%7B%5BN%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
. This can easily be seen to be a norm on functions ![{U^2[N]}](https://s0.wp.com/latex.php?latex=%7BU%5E2%5BN%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
norm for 
, then we can write 
with a residue class modulo 
, and from Fourier expansion we have ![\displaystyle \mathop{\bf E}_{n \in [N]} f 1_P(n) \ll \sup_\alpha |\mathop{\bf E}_{n \in [N]} f 1_I(n) e(\alpha n)|.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+f+1_P%28n%29+%5Cll+%5Csup_%5Calpha+%7C%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+f+1_I%28n%29+e%28%5Calpha+n%29%7C.&bg=ffffff&fg=000000&s=0&c=20201002)
be a standard bump function supported on ![{[-1,1]}](https://s0.wp.com/latex.php?latex=%7B%5B-1%2C1%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
with total mass and 
is a parameter then ![\displaystyle \mathop{\bf E}_{n \in [N]} f 1_I(n) e(\alpha n)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+f+1_I%28n%29+e%28%5Calpha+n%29+&bg=ffffff&fg=000000&s=0&c=20201002)
![\displaystyle \ll |\mathop{\bf E}_{n \in [-N,2N]; h, k \in [-N,N]} \frac{1}{\delta} \psi(\frac{h}{\delta N})](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5Cll+%7C%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5B-N%2C2N%5D%3B+h%2C+k+%5Cin+%5B-N%2CN%5D%7D+%5Cfrac%7B1%7D%7B%5Cdelta%7D+%5Cpsi%28%5Cfrac%7Bh%7D%7B%5Cdelta+N%7D%29&bg=ffffff&fg=000000&s=0&c=20201002)
![\displaystyle \ll |\mathop{\bf E}_{n \in [-N,2N]; h, k \in [-N,N]} \frac{1}{\delta} \psi(\frac{h}{\delta N}) 1_I(n+k) f(n+h+k) e(\alpha(n+h+k))|](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cll+%7C%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5B-N%2C2N%5D%3B+h%2C+k+%5Cin+%5B-N%2CN%5D%7D+%5Cfrac%7B1%7D%7B%5Cdelta%7D+%5Cpsi%28%5Cfrac%7Bh%7D%7B%5Cdelta+N%7D%29+1_I%28n%2Bk%29+f%28n%2Bh%2Bk%29+e%28%5Calpha%28n%2Bh%2Bk%29%29%7C+&bg=ffffff&fg=000000&s=0&c=20201002)
![\displaystyle \mathop{\bf E}_{h \in [-N,N]} \frac{1}{\delta} \psi(\frac{h}{\delta N}) 1_I(n+h+k) - \mathop{\bf E}_{h \in [-N,N]} \frac{1}{\delta} \psi(\frac{h}{\delta N}) 1_I(n+k)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathop%7B%5Cbf+E%7D_%7Bh+%5Cin+%5B-N%2CN%5D%7D+%5Cfrac%7B1%7D%7B%5Cdelta%7D+%5Cpsi%28%5Cfrac%7Bh%7D%7B%5Cdelta+N%7D%29+1_I%28n%2Bh%2Bk%29+-+%5Cmathop%7B%5Cbf+E%7D_%7Bh+%5Cin+%5B-N%2CN%5D%7D+%5Cfrac%7B1%7D%7B%5Cdelta%7D+%5Cpsi%28%5Cfrac%7Bh%7D%7B%5Cdelta+N%7D%29+1_I%28n%2Bk%29&bg=ffffff&fg=000000&s=0&c=20201002)
we now have ![\displaystyle \mathop{\bf E}_{n \in [N]} f 1_P(n) \ll \frac{1}{\delta} \sup_{\alpha,\beta} |\mathop{\bf E}_{n \in [N]; h, k \in [-N,N]} e(\beta h + \alpha (n+h+k))](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+f+1_P%28n%29+%5Cll+%5Cfrac%7B1%7D%7B%5Cdelta%7D+%5Csup_%7B%5Calpha%2C%5Cbeta%7D+%7C%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%3B+h%2C+k+%5Cin+%5B-N%2CN%5D%7D+e%28%5Cbeta+h+%2B+%5Calpha+%28n%2Bh%2Bk%29%29+&bg=ffffff&fg=000000&s=0&c=20201002)
as a linear combination of 
and using the Gowers–Cauchy–Schwarz inequality, we conclude ![\displaystyle \mathop{\bf E}_{n \in [N]} f 1_P(n) \ll \frac{1}{\delta} \|f\|_{U^2([N])} + \delta](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+f+1_P%28n%29+%5Cll+%5Cfrac%7B1%7D%7B%5Cdelta%7D+%5C%7Cf%5C%7C_%7BU%5E2%28%5BN%5D%29%7D+%2B+%5Cdelta&bg=ffffff&fg=000000&s=0&c=20201002)
we have ![\displaystyle \| f\|_{U^{1^+}[N]} \ll \|f\|_{U^2[N]}^{1/2}.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5C%7C+f%5C%7C_%7BU%5E%7B1%5E%2B%7D%5BN%5D%7D+%5Cll+%5C%7Cf%5C%7C_%7BU%5E2%5BN%5D%7D%5E%7B1%2F2%7D.&bg=ffffff&fg=000000&s=0&c=20201002)
with this insertion.
had true complexity 
one needs to localize the 
. But I will ignore this technical point to keep the exposition simple.] The weaker claim that 
has true complexity 
for some 
, then there is a Fourier phase 
such that ![\displaystyle |\mathop{\bf E}_{n \in [N]} f(n) e(-\alpha n)| \gg \eta^2;](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7C%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+f%28n%29+e%28-%5Calpha+n%29%7C+%5Cgg+%5Ceta%5E2%3B&bg=ffffff&fg=000000&s=0&c=20201002)
![\displaystyle |\mathop{\bf E}_{n \in [N]} f(n) e(-\alpha n)| \ll \|f\|_{U^2[N]}.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7C%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+f%28n%29+e%28-%5Calpha+n%29%7C+%5Cll+%5C%7Cf%5C%7C_%7BU%5E2%5BN%5D%7D.&bg=ffffff&fg=000000&s=0&c=20201002)
![\displaystyle |\mathop{\bf E}_{n \in [N]} f(n)| \geq \eta.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7C%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+f%28n%29%7C+%5Cgeq+%5Ceta.&bg=ffffff&fg=000000&s=0&c=20201002)
appearing in the ![{\|f\|_{U^{1^+}[N]} \geq \eta}](https://s0.wp.com/latex.php?latex=%7B%5C%7Cf%5C%7C_%7BU%5E%7B1%5E%2B%7D%5BN%5D%7D+%5Cgeq+%5Ceta%7D&bg=ffffff&fg=000000&s=0&c=20201002)
, thus ![\displaystyle |\mathop{\bf E}_{n \in [N]} 1_P(n) f(n)| \geq \eta](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7C%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+1_P%28n%29+f%28n%29%7C+%5Cgeq+%5Ceta&bg=ffffff&fg=000000&s=0&c=20201002)
. We write the above inequality as ![\displaystyle |\mathop{\bf E}_{n \in [N]} 1_{n=b\ (q)} 1_I(n) f(n)| \geq \eta](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7C%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+1_%7Bn%3Db%5C+%28q%29%7D+1_I%28n%29+f%28n%29%7C+%5Cgeq+%5Ceta&bg=ffffff&fg=000000&s=0&c=20201002)
and some interval ![\displaystyle |\mathop{\bf E}_{n \in [N]} e(-an/q) 1_I(n) f(n)| \geq \eta](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7C%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+e%28-an%2Fq%29+1_I%28n%29+f%28n%29%7C+%5Cgeq+%5Ceta&bg=ffffff&fg=000000&s=0&c=20201002)
. Convolving 
by 
for ![\displaystyle |\mathop{\bf E}_{n \in [N]} e(-an/q) (1_I * \psi_\delta)(n) f(n)| \gg \eta.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7C%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+e%28-an%2Fq%29+%281_I+%2A+%5Cpsi_%5Cdelta%29%28n%29+f%28n%29%7C+%5Cgg+%5Ceta.&bg=ffffff&fg=000000&s=0&c=20201002)
of 
is bounded by 
and supported on ![{[-\frac{1}{\delta N},\frac{1}{\delta N}]}](https://s0.wp.com/latex.php?latex=%7B%5B-%5Cfrac%7B1%7D%7B%5Cdelta+N%7D%2C%5Cfrac%7B1%7D%7B%5Cdelta+N%7D%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
, thus by Fourier expansion and the triangle inequality we have ![\displaystyle |\mathop{\bf E}_{n \in [N]} e(-an/q) e(-\xi n) f(n)| \gg \eta^2](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7C%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+e%28-an%2Fq%29+e%28-%5Cxi+n%29+f%28n%29%7C+%5Cgg+%5Ceta%5E2&bg=ffffff&fg=000000&s=0&c=20201002)
![{\xi \in [-\frac{1}{\delta N},\frac{1}{\delta N}]}](https://s0.wp.com/latex.php?latex=%7B%5Cxi+%5Cin+%5B-%5Cfrac%7B1%7D%7B%5Cdelta+N%7D%2C%5Cfrac%7B1%7D%7B%5Cdelta+N%7D%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
, so in particular 
. Thus we have ![\displaystyle |\mathop{\bf E}_{n \in [N]} f(n) e(-\alpha n)| \gg \eta^2 \ \ \ \ \ (2)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7C%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+f%28n%29+e%28-%5Calpha+n%29%7C+%5Cgg+%5Ceta%5E2+%5C+%5C+%5C+%5C+%5C+%282%29&bg=ffffff&fg=000000&s=0&c=20201002)
with 
. Conversely, for ![\displaystyle |\mathop{\bf E}_{n \in [N]} f(n) e(-\alpha n)| \ll \frac{q}{\eta} \|f\|_{U^{1^+}[N]} \ll \frac{1}{\eta^2} \|f\|_{U^{1^+}[N]}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%7C%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+f%28n%29+e%28-%5Calpha+n%29%7C+%5Cll+%5Cfrac%7Bq%7D%7B%5Ceta%7D+%5C%7Cf%5C%7C_%7BU%5E%7B1%5E%2B%7D%5BN%5D%7D+%5Cll+%5Cfrac%7B1%7D%7B%5Ceta%5E2%7D+%5C%7Cf%5C%7C_%7BU%5E%7B1%5E%2B%7D%5BN%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
![{\|f\|_{U^{1^+}[N]} \gg \eta^4}](https://s0.wp.com/latex.php?latex=%7B%5C%7Cf%5C%7C_%7BU%5E%7B1%5E%2B%7D%5BN%5D%7D+%5Cgg+%5Ceta%5E4%7D&bg=ffffff&fg=000000&s=0&c=20201002)
.
of functions (where each class of functions ![{\| f \|_{{\mathcal F}^*} := \sup_{\phi \in {\mathcal F}} \mathop{\bf E}_{n \in[N]} f(n) \overline{\phi(n)}}](https://s0.wp.com/latex.php?latex=%7B%5C%7C+f+%5C%7C_%7B%7B%5Cmathcal+F%7D%5E%2A%7D+%3A%3D+%5Csup_%7B%5Cphi+%5Cin+%7B%5Cmathcal+F%7D%7D+%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin%5BN%5D%7D+f%28n%29+%5Coverline%7B%5Cphi%28n%29%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
:
inverse theorem, namely degree two nilsequences, bracket quadratic phases, and local quadratic phases, as well as the more narrow class of globally quadratic phases.
, known as Host-Kra seminorms. The 
norm can be defined for ![\displaystyle \|f\|_{U^1(X)} := \lim_{N \rightarrow \infty} \int_X |\mathop{\bf E}_{n \in [N]} T^n f|\ d\mu](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5C%7Cf%5C%7C_%7BU%5E1%28X%29%7D+%3A%3D+%5Clim_%7BN+%5Crightarrow+%5Cinfty%7D+%5Cint_X+%7C%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+T%5En+f%7C%5C+d%5Cmu&bg=ffffff&fg=000000&s=0&c=20201002)
![\displaystyle \|f\|_{U^2(X)}^4 := \lim_{N \rightarrow \infty} \mathop{\bf E}_{n \in [N]} \| T^n f \overline{f} \|_{U^1(X)}^2.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5C%7Cf%5C%7C_%7BU%5E2%28X%29%7D%5E4+%3A%3D+%5Clim_%7BN+%5Crightarrow+%5Cinfty%7D+%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+%5C%7C+T%5En+f+%5Coverline%7Bf%7D+%5C%7C_%7BU%5E1%28X%29%7D%5E2.&bg=ffffff&fg=000000&s=0&c=20201002)
(generated by the (almost everywhere) invariant measurable subsets of 
, generated by the eigenfunctions of 
for some ![{U^{1^+}[N]}](https://s0.wp.com/latex.php?latex=%7BU%5E%7B1%5E%2B%7D%5BN%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
norm, one can then define the Host-Kra 
seminorm by ![\displaystyle \|f\|_{U^{1^+}(X)} := \sup_{q \geq 1} \frac{1}{q} \lim_{N \rightarrow \infty} \int_X |\mathop{\bf E}_{n \in [N]} T^{qn} f|\ d\mu;](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5C%7Cf%5C%7C_%7BU%5E%7B1%5E%2B%7D%28X%29%7D+%3A%3D+%5Csup_%7Bq+%5Cgeq+1%7D+%5Cfrac%7B1%7D%7Bq%7D+%5Clim_%7BN+%5Crightarrow+%5Cinfty%7D+%5Cint_X+%7C%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+T%5E%7Bqn%7D+f%7C%5C+d%5Cmu%3B&bg=ffffff&fg=000000&s=0&c=20201002)
, generated by the periodic sets of ![{\mathop{\bf E}_{n \in [N]}}](https://s0.wp.com/latex.php?latex=%7B%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
and the exponential notation 
. The prime number theorem in qualitative form asserts that ![\displaystyle \mathop{\bf E}_{n \in [N]} \mu(n) = o(1)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+%5Cmu%28n%29+%3D+o%281%29&bg=ffffff&fg=000000&s=0&c=20201002)
. With Vinogradov-Korobov error term, the prime number theorem is strengthened to ![\displaystyle \mathop{\bf E}_{n \in [N]} \mu(n) \ll \exp( - c \log^{3/5} N (\log \log N)^{-1/5} );](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+%5Cmu%28n%29+%5Cll+%5Cexp%28+-+c+%5Clog%5E%7B3%2F5%7D+N+%28%5Clog+%5Clog+N%29%5E%7B-1%2F5%7D+%29%3B&bg=ffffff&fg=000000&s=0&c=20201002)
type factors) as pseudopolynomial decay. Equivalently, we obtain pseudopolynomial decay of Gowers ![\displaystyle \| \mu \|_{U^1([N])} \ll \exp( - c \log^{3/5} N (\log \log N)^{-1/5} ).](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5C%7C+%5Cmu+%5C%7C_%7BU%5E1%28%5BN%5D%29%7D+%5Cll+%5Cexp%28+-+c+%5Clog%5E%7B3%2F5%7D+N+%28%5Clog+%5Clog+N%29%5E%7B-1%2F5%7D+%29.&bg=ffffff&fg=000000&s=0&c=20201002)
![\displaystyle \| \mu \|_{U^1([N])} \ll_\varepsilon N^{-1/2+\varepsilon}](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5C%7C+%5Cmu+%5C%7C_%7BU%5E1%28%5BN%5D%29%7D+%5Cll_%5Cvarepsilon+N%5E%7B-1%2F2%2B%5Cvarepsilon%7D&bg=ffffff&fg=000000&s=0&c=20201002)
![\displaystyle \| \mu 1_{a \hbox{ mod } q}\|_{U^1([N])} \ll_A \log^{-A} N \ \ \ \ \ (1)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5C%7C+%5Cmu+1_%7Ba+%5Chbox%7B+mod+%7D+q%7D%5C%7C_%7BU%5E1%28%5BN%5D%29%7D+%5Cll_A+%5Clog%5E%7B-A%7D+N+%5C+%5C+%5C+%5C+%5C+%281%29&bg=ffffff&fg=000000&s=0&c=20201002)
and any 
, but with the catch that the implied constant is ![\displaystyle \mathop{\bf E}_{n \in [N]} \mu(n) e(-\alpha n) \ll_A \log^{-A} N](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+%5Cmu%28n%29+e%28-%5Calpha+n%29+%5Cll_A+%5Clog%5E%7B-A%7D+N&bg=ffffff&fg=000000&s=0&c=20201002)
, which leads (by standard Fourier arguments) to the Fourier uniformity estimate ![\displaystyle \| \mu \|_{U^2([N])} \ll_A \log^{-A} N.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5C%7C+%5Cmu+%5C%7C_%7BU%5E2%28%5BN%5D%29%7D+%5Cll_A+%5Clog%5E%7B-A%7D+N.&bg=ffffff&fg=000000&s=0&c=20201002)
![\displaystyle \| \mu \|_{U^2([N])} \ll \log^{-c} N \ \ \ \ \ (2)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5C%7C+%5Cmu+%5C%7C_%7BU%5E2%28%5BN%5D%29%7D+%5Cll+%5Clog%5E%7B-c%7D+N+%5C+%5C+%5C+%5C+%5C+%282%29&bg=ffffff&fg=000000&s=0&c=20201002)
.
![\displaystyle \mathop{\bf E}_{n \in [N]} \mu(n) \overline{F}(g(n) \Gamma) \ll_{A,F,G/\Gamma} \log^{-A} N \ \ \ \ \ (3)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+%5Cmu%28n%29+%5Coverline%7BF%7D%28g%28n%29+%5CGamma%29+%5Cll_%7BA%2CF%2CG%2F%5CGamma%7D+%5Clog%5E%7B-A%7D+N+%5C+%5C+%5C+%5C+%5C+%283%29&bg=ffffff&fg=000000&s=0&c=20201002)
, any polynomial sequence 
, and any Lipschitz function ![\displaystyle \| \mu \|_{U^3([N])} \geq \delta](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5C%7C+%5Cmu+%5C%7C_%7BU%5E3%28%5BN%5D%29%7D+%5Cgeq+%5Cdelta&bg=ffffff&fg=000000&s=0&c=20201002)
, then there exists a degree two nilmanifold 
, complexity 
such that ![\displaystyle \mathop{\bf E}_{n \in [N]} \mu(n) \overline{F}(g(n) \Gamma) \gg \exp(-\delta^{-O(1)}).](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+%5Cmu%28n%29+%5Coverline%7BF%7D%28g%28n%29+%5CGamma%29+%5Cgg+%5Cexp%28-%5Cdelta%5E%7B-O%281%29%7D%29.&bg=ffffff&fg=000000&s=0&c=20201002)
![\displaystyle \| \mu \|_{U^3([N])} = o(1).](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5C%7C+%5Cmu+%5C%7C_%7BU%5E3%28%5BN%5D%29%7D+%3D+o%281%29.&bg=ffffff&fg=000000&s=0&c=20201002)
produced by this argument is completely ineffective: obtaining a bound on when this 
, the situation is even worse, because the quantitative inverse theory for these norms is poorer, and indeed it was only with the 
). Basically, Manners establishes if ![\displaystyle \| \mu \|_{U^k([N])} \geq \delta](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5C%7C+%5Cmu+%5C%7C_%7BU%5Ek%28%5BN%5D%29%7D+%5Cgeq+%5Cdelta&bg=ffffff&fg=000000&s=0&c=20201002)
nilmanifold 
, a polynomial sequence 
such that ![\displaystyle \mathop{\bf E}_{n \in [N]} \mu(n) \overline{F}(g(n) \Gamma) \gg \exp\exp(-\delta^{-O(1)}).](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+%5Cmu%28n%29+%5Coverline%7BF%7D%28g%28n%29+%5CGamma%29+%5Cgg+%5Cexp%5Cexp%28-%5Cdelta%5E%7B-O%281%29%7D%29.&bg=ffffff&fg=000000&s=0&c=20201002)
![\displaystyle \| \mu \|_{U^k([N])} = o(1)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5C%7C+%5Cmu+%5C%7C_%7BU%5Ek%28%5BN%5D%29%7D+%3D+o%281%29&bg=ffffff&fg=000000&s=0&c=20201002)
, we have ![{\| \mu \|_{U^k([N])} \ll (\log\log N)^{-c_k}}](https://s0.wp.com/latex.php?latex=%7B%5C%7C+%5Cmu+%5C%7C_%7BU%5Ek%28%5BN%5D%29%7D+%5Cll+%28%5Clog%5Clog+N%29%5E%7B-c_k%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
for some 
. The implied constants are effective. 
case there is some hope to remove this logarithm based on the improved quantitative inverse theory currently available in this case, but there is a technical obstruction to doing so which we will discuss later in this post. For 
the above bound is the best one could hope to achieve purely using the quantitative inverse theory of Manners.
to ![\displaystyle \| \Lambda - 1 \|_{U^1([N])} \ll \exp( - c \log^{3/5} N (\log \log N)^{-1/5} )](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5C%7C+%5CLambda+-+1+%5C%7C_%7BU%5E1%28%5BN%5D%29%7D+%5Cll+%5Cexp%28+-+c+%5Clog%5E%7B3%2F5%7D+N+%28%5Clog+%5Clog+N%29%5E%7B-1%2F5%7D+%29&bg=ffffff&fg=000000&s=0&c=20201002)
is the quasipolynomial quantity 
![\displaystyle \mathop{\bf E}_{n \in [N]} (\Lambda - \Lambda_{\hbox{Cram\'er}}(n)) e(-\alpha n) \ll_A \log^{-A} N](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+%28%5CLambda+-+%5CLambda_%7B%5Chbox%7BCram%5C%27er%7D%7D%28n%29%29+e%28-%5Calpha+n%29+%5Cll_A+%5Clog%5E%7B-A%7D+N&bg=ffffff&fg=000000&s=0&c=20201002)
![\displaystyle \| \Lambda - \Lambda_{\hbox{Cram\'er}}\|_{U^2([N])} \ll_A \log^{-A} N](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5C%7C+%5CLambda+-+%5CLambda_%7B%5Chbox%7BCram%5C%27er%7D%7D%5C%7C_%7BU%5E2%28%5BN%5D%29%7D+%5Cll_A+%5Clog%5E%7B-A%7D+N&bg=ffffff&fg=000000&s=0&c=20201002)
![{\| \Lambda - \Lambda_{\hbox{Cram\'er}} \|_{U^k([N])} \ll (\log\log N)^{-c_k}}](https://s0.wp.com/latex.php?latex=%7B%5C%7C+%5CLambda+-+%5CLambda_%7B%5Chbox%7BCram%5C%27er%7D%7D+%5C%7C_%7BU%5Ek%28%5BN%5D%29%7D+%5Cll+%28%5Clog%5Clog+N%29%5E%7B-c_k%7D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
for some 
with respect to the main term.
with a zero 
with 
for some small absolute constant ![\displaystyle \| \mu 1_{a \hbox{ mod } q}\|_{U^1([N])} \ll \exp(-\log^c N). \ \ \ \ \ (5)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5C%7C+%5Cmu+1_%7Ba+%5Chbox%7B+mod+%7D+q%7D%5C%7C_%7BU%5E1%28%5BN%5D%29%7D+%5Cll+%5Cexp%28-%5Clog%5Ec+N%29.+%5C+%5C+%5C+%5C+%5C+%285%29&bg=ffffff&fg=000000&s=0&c=20201002)
![\displaystyle \mathop{\bf E}_{n \in [N]} \mu(n) \overline{F}(g(n) \Gamma) \ll \exp(-\log^c N) \ \ \ \ \ (6)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+%5Cmu%28n%29+%5Coverline%7BF%7D%28g%28n%29+%5CGamma%29+%5Cll+%5Cexp%28-%5Clog%5Ec+N%29+%5C+%5C+%5C+%5C+%5C+%286%29&bg=ffffff&fg=000000&s=0&c=20201002)
and complexity 
, all polynomial sequences 
that takes the Siegel zero into account, and in particular is such that one has the (effective) pseudopolynomial bound ![\displaystyle \| (\mu - \mu_{\hbox{Siegel}}) 1_{a \hbox{ mod } q}\|_{U^1([N])} \ll \exp(-\log^c N) \ \ \ \ \ (7)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5C%7C+%28%5Cmu+-+%5Cmu_%7B%5Chbox%7BSiegel%7D%7D%29+1_%7Ba+%5Chbox%7B+mod+%7D+q%7D%5C%7C_%7BU%5E1%28%5BN%5D%29%7D+%5Cll+%5Cexp%28-%5Clog%5Ec+N%29+%5C+%5C+%5C+%5C+%5C+%287%29&bg=ffffff&fg=000000&s=0&c=20201002)
, and 
, and 
and plays only a minor role in the analysis). This is a rather complicated formula, but it seems to be virtually the only choice of approximant that allows for bounds such as ![\displaystyle \mathop{\bf E}_{n \in [N]} (\mu - \mu_{\hbox{Siegel}})(n) \overline{F}(g(n) \Gamma) \ll \exp(-\log^c N)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+%28%5Cmu+-+%5Cmu_%7B%5Chbox%7BSiegel%7D%7D%29%28n%29+%5Coverline%7BF%7D%28g%28n%29+%5CGamma%29+%5Cll+%5Cexp%28-%5Clog%5Ec+N%29+&bg=ffffff&fg=000000&s=0&c=20201002)
as before, which when combined with Manners’ inverse theorem gives the doubly logarithmic bound ![\displaystyle \| \mu - \mu_{\hbox{Siegel}} \|_{U^k([N])} \ll (\log\log N)^{-c_k}.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5C%7C+%5Cmu+-+%5Cmu_%7B%5Chbox%7BSiegel%7D%7D+%5C%7C_%7BU%5Ek%28%5BN%5D%29%7D+%5Cll+%28%5Clog%5Clog+N%29%5E%7B-c_k%7D.&bg=ffffff&fg=000000&s=0&c=20201002)
![\displaystyle \| \mu_{\hbox{Siegel}} \|_{U^k([N])} \ll \log^{-c_k} N](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5C%7C+%5Cmu_%7B%5Chbox%7BSiegel%7D%7D+%5C%7C_%7BU%5Ek%28%5BN%5D%29%7D+%5Cll+%5Clog%5E%7B-c_k%7D+N&bg=ffffff&fg=000000&s=0&c=20201002)
is simpler; we ended up using 
![\displaystyle \| (\Lambda - \Lambda_{\hbox{Siegel}}) 1_{a \hbox{ mod } q}\|_{U^1([N])} \ll \exp(-\log^c N).](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5C%7C+%28%5CLambda+-+%5CLambda_%7B%5Chbox%7BSiegel%7D%7D%29+1_%7Ba+%5Chbox%7B+mod+%7D+q%7D%5C%7C_%7BU%5E1%28%5BN%5D%29%7D+%5Cll+%5Cexp%28-%5Clog%5Ec+N%29.&bg=ffffff&fg=000000&s=0&c=20201002)
![\displaystyle \mathop{\bf E}_{n \in [N]} (\Lambda - \Lambda_{\hbox{Siegel}})(n) \overline{F}(g(n) \Gamma) \ll \exp(-\log^c N) \ \ \ \ \ (8)](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5Cmathop%7B%5Cbf+E%7D_%7Bn+%5Cin+%5BN%5D%7D+%28%5CLambda+-+%5CLambda_%7B%5Chbox%7BSiegel%7D%7D%29%28n%29+%5Coverline%7BF%7D%28g%28n%29+%5CGamma%29+%5Cll+%5Cexp%28-%5Clog%5Ec+N%29+%5C+%5C+%5C+%5C+%5C+%288%29&bg=ffffff&fg=000000&s=0&c=20201002)
![\displaystyle \| \Lambda_{\hbox{Siegel}} \|_{U^k([N])} \ll \log^{-c_k} N.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5C%7C+%5CLambda_%7B%5Chbox%7BSiegel%7D%7D+%5C%7C_%7BU%5Ek%28%5BN%5D%29%7D+%5Cll+%5Clog%5E%7B-c_k%7D+N.&bg=ffffff&fg=000000&s=0&c=20201002)
![\displaystyle \| \Lambda - \Lambda_{\hbox{Siegel}} \|_{U^k([N])} \ll (\log\log N)^{-c_k}.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%5C%7C+%5CLambda+-+%5CLambda_%7B%5Chbox%7BSiegel%7D%7D+%5C%7C_%7BU%5Ek%28%5BN%5D%29%7D+%5Cll+%28%5Clog%5Clog+N%29%5E%7B-c_k%7D.&bg=ffffff&fg=000000&s=0&c=20201002)
that fails to be 
(up to errors of 
), where the coefficients are of size 
; current methods only give coefficient bounds of the form 
. An 
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