Joni Teräväinen and I have just uploaded to the arXiv our preprint “The Hardy–Littlewood–Chowla conjecture in the presence of a Siegel zero“. This paper is a development of the theme that certain conjectures in analytic number theory become easier if one makes the hypothesis that Siegel zeroes exist; this places one in a presumably “illusory” universe, since the widely believed Generalised Riemann Hypothesis (GRH) precludes the existence of such zeroes, yet this illusory universe seems remarkably self-consistent and notoriously impossible to eliminate from one’s analysis.
For the purposes of this paper, a Siegel zero is a zero 
(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 
to “pretend” to be like the exceptional character for primes of magnitude comparable to . Indeed, if we define an exceptional prime to be a prime 
in which 
, then very few primes near will be exceptional; in our paper we use some elementary arguments to establish the bounds 
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 this blog post for some variants of this argument, which were inspired by work of Heath-Brown.) There is also a companion bound (somewhat weaker) that covers a range of a little bit below 
.
One of the early influential results in this area was the following result of Heath-Brown, which I previously blogged about here:
Theorem 1 (Hardy-Littlewood assuming Siegel zero) Letbe a fixed natural number. Suppose one has a Siegel zero
for all
, where
is the von Mangoldt function and
is the singular series
In particular, Heath-Brown showed that if there are infinitely many Siegel zeroes, then there are also infinitely many twin primes, with the correct asymptotic predicted by the Hardy-Littlewood prime tuple conjecture at infinitely many scales.
Very recently, Chinis established an analogous result for the Chowla conjecture (building upon earlier work of Germán and Katai):
Theorem 2 (Chowla assuming Siegel zero) Letbe distinct fixed natural numbers. Suppose one has a Siegel zero
in the range
, where
In our paper we unify these results and also improve the quantitative estimates and range of
Theorem 3 (Hardy-Littlewood-Chowla assuming Siegel zero) Letbe distinct fixed natural numbers with
. Suppose one has a Siegel zero
for
for any fixed
Our argument proceeds by a series of steps in which we replace
- (i) Replace the Liouville function
, which is a completely multiplicative function that agrees with
- (ii) Replace the von Mangoldt function
, which is the Dirichlet convolution
multiplied by a Selberg sieve weight
to essentially restrict that convolution to almost primes.
- (iii) Replace
which has the structure of a “Type I sum”, and which agrees with
- (iv) Replace the approximant
which has the structure of a “Type I sum”.
- (v) Now that all terms in the correlation have been replaced with tractable Type I sums, use standard Euler product calculations and Fourier analysis, similar in spirit to the proof of the pseudorandomness of the Selberg sieve majorant for the primes in this paper of Ben Green and myself, to evaluate the correlation to high accuracy.
Steps (i), (ii) proceed mainly through estimates such as (1) and standard sieve theory bounds. Step (iii) is based primarily on estimates on the number of smooth numbers of a certain size.
The restriction 
this sum can be easily controlled by the Dirichlet hyperbola method. For 
one needs the fact that has a level of distribution greater than 
; 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 . Unfortunately for 
there are no known techniques to unconditionally obtain asymptotics, even for the model sum 
hypothesis in our main theorem at our current level of understanding of analytic number theory.
Step (v) is a tedious but straightforward sieve theoretic computation, similar in many ways to the correlation estimates of Goldston and Yildirim used in their work on small gaps between primes (as discussed for instance here), and then also used by Ben Green and myself to locate arithmetic progressions in primes.
which is nowhere analytic, but is such that the Taylor series 
converges for every 
? (Of course, this series would not then converge to 
, but instead to some analytic function 
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 
, a (periodic) subset 
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 
-dimensional projective variety 
there is a standard line bundle 
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 
. Prior to this paper, the best such constructions of 
given 
given by 
; the method also gives comparable bounds for some other related algebraic geometry statistics, such as the largest 
for which the pluricanonical map associated to the linear system 
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 
constant to 
.
![{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 
. Thus for instance ![\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 
. Let us informally say that a form 
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 
such that ![{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 
).
norms, which I will call the 
(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 
as ![\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 
norm is orthogonal to the Kronecker factor 
, 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)
.) Meanwhile, the bound ![\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 
, there are at most 
. Currently, the largest number of solutions that is known to be attainable is eight, with 
equal to 
of Pascal’s triangle it is natural to restrict attention to the left half 
of the triangle.
and 
, there are at most two solutions to 
); we have deleted the 
case as it of course automatically supplies exactly one solution to 
for 
in the region 
, 
and 
is the 
Fibonacci number.
in Theorem 
primarily as real numbers rather than integers, and express 
as 
whose asymptotics are easily computable (for instance one has the asymptotic 
). One can then view the problem as one of trying to control the number of lattice points on the graph 
. Here we can take advantage of the fact that in the regime 
(which corresponds to working in the left half 
of Pascal’s triangle), the function 
). This can be used to preclude the possibility of having a cluster of three or more nearby lattice points on the graph 
is a solution to 
to this equation in the left half with 
to 
-adic valuations
of the binomial coefficients, defined as the number of times a prime 
. From the fundamental theorem of arithmetic, a collision 
denotes the fractional part of 
is larger than 
.)
![{[P, P + P \log^{-100} P]}](https://s0.wp.com/latex.php?latex=%7B%5BP%2C+P+%2B+P+%5Clog%5E%7B-100%7D+P%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
for some suitably chosen scale parameter 
are far apart from each other, then one would expect the left-hand side of 
, since this term shows up in the summation on the left-hand side but not the right. Similarly if 
are far apart from each other (although there are some annoying cases one has to treat separately when there is some “unexpected commensurability”, for instance if 
is a rational multiple of 
, where 
. Fortunately, the methods of Vinogradov (which more generally can handle sums such as 
and 
for various analytic functions 
and 
are not too large compared to 
, and this ultimately leads to a distance bound 
is the 
of groups, then one of three things has to happen:
of 
of 
. 
.
into a group 
such that 
for all 
. 
of 
be the 
to be the quotient group 
and 
of some subgroup 
of 
for some non-trivial normal subgroup 
is the quotient map. 
.
such that 
for all 
.
![\displaystyle G_2 := [G,G], G_3 := [G,G_2], \dots](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++G_2+%3A%3D+%5BG%2CG%5D%2C+G_3+%3A%3D+%5BG%2CG_2%5D%2C+%5Cdots&bg=ffffff&fg=000000&s=0&c=20201002)
is a normal subgroup of 
is also a subgroup of 
is a non-trivial homomorphism to an abelian group 
that annihilates 
, and thus ![\displaystyle G_2 = [G,G] = [G, HG_2].](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++G_2+%3D+%5BG%2CG%5D+%3D+%5BG%2C+HG_2%5D.&bg=ffffff&fg=000000&s=0&c=20201002)
, ![\displaystyle [G, HG_2] \subset [G,H] G_3 \subset H G_3](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%5BG%2C+HG_2%5D+%5Csubset+%5BG%2CH%5D+G_3+%5Csubset+H+G_3&bg=ffffff&fg=000000&s=0&c=20201002)
. One can continue this argument by induction to show that 
for every 
; taking 
. Thus 
be a prime, and let 
, where 
is the ring of 
such that 
for all 
.
there exists 
, which we write as 
and 
with 
; taking limits as 
using the closed nature of 
is already handled, so now suppose 
. If 
where 
. Thus to establish the 
.
for some 
with 
. By the 
for some 
. Raising to the 
power and using 
and 
, we note that 
matrix of trace zero with coefficients in 
is a linear combination of 
, 
, 
and is thus a sum of matrices that square to zero. Hence, if 
, then 
for some matrix 
of trace zero, and thus one can write 
errors) as the finite product of matrices of the form 
with 
. By the previous arguments, such a matrix 
lies in 
and the claim has already been proven for 
. Arguing as before, it suffices to close the induction under the additional hypothesis that 
, thus we may write 
. By induction hypothesis, we may find 
. But then 
, and we are done. 
rather than just one:
of two nilpotent groups 
. Then exactly one of the following hold:
, 
into an abelian group 
non-trivial, such that 
for all 
, where 
is the projection of 
. 
for some subgroup 
of 
. This is a subgroup of 
whenever 
. For any 
in the projection 
such that 
whenever 
is a homomorphism, so we are in option (i).
of an element 
. Then exactly one of the following hold:
such that 
.
such that 
. Similarly, there exists 
with 
. Taking commutators of 
, we can find 
with 
. Continuing to take commutators with 
with 
. Thus, the closed subgroup 
of 
. Similarly 
; multiplying, we end up in conclusion (ii). 
for some subgroups 
of 
or 
(or both). 
of 
, where 
is the quotient map. 
such that 
is the graph of 
. In particular, 
, 
from 
and 
fail to be surjective, then we are in option (i). Thus we may assume that these maps are surjective.
, 
is non-trivial. We can then descend down to the quotient 
are both bijections, hence are group isomorphisms. If we set 
we end up in case (iii). 
and a non-trivial group homomorphism 
such that 
for all 
that is known in ergodic theory contexts as the “Furstenberg-Weiss argument”, as an argument of this type arose in 
of three groups 
. Then one of the following hold:
of 
whenever 
and 
. 
of 
abelian, such that 
, where 
is the quotient map. 
is a proper subgroup of 
), so we may assume that 
be any two elements of 
such that 
terms, we conclude that ![\displaystyle (1, 1, [g_3,g'_3]) \in H.](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle++%281%2C+1%2C+%5Bg_3%2Cg%27_3%5D%29+%5Cin+H.&bg=ffffff&fg=000000&s=0&c=20201002)
contains every commutator ![{[g_3,g'_3]}](https://s0.wp.com/latex.php?latex=%7B%5Bg_3%2Cg%27_3%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
, and thus contains the entire group ![{[G_3,G_3]}](https://s0.wp.com/latex.php?latex=%7B%5BG_3%2CG_3%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
generated by these commutators. If ![{G_1 \times G_2 \times G_3/[G_3,G_3]}](https://s0.wp.com/latex.php?latex=%7BG_1+%5Ctimes+G_2+%5Ctimes+G_3%2F%5BG_3%2CG_3%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
in the obvious fashion, placing one in option (ii). Hence the only remaining case is when 
, 
for some abelian group 
non-trivial, such that 
whenever 
is the projection of 
. 
is a variable ranging in some subgroup 
in terms of either 
is not of the form 
, since it only contains those 
with 
, with 
whenever 
of 
for some 
such that 
whenever 
of 
, where 
is the quotient map. 
, 
, 
to an abelian group 
is then a normal subgroup of 
for some map 
. As 
for some homomorphisms 
, 
. Thus elements 
.
or 
fails to be injective, then we can quotient out by their kernels and end up in option (ii). If 
also fails to be surjective (since for 
is now constrained to lie in the range of 
are bijective and thus group isomorphisms. Setting 
to the identity, we arrive at option (iii). 
to some abelian group 
for some 
not both trivial, such that 
whenever 
such that 
. 
. 
.
. We can assume that each ![{G_i = [G_i,G_i]}](https://s0.wp.com/latex.php?latex=%7BG_i+%3D+%5BG_i%2CG_i%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
, otherwise we can quotient out by the commutator and arrive in option (iii). Similarly, we may assume that all the projections of 
and any 
, one can find 
for 
and 
. For 
. Now suppose inductively that 
and the claim has already been proven for 
is perfect, it suffices to establish this claim for 
of the form ![{g_k = [g'_k, g''_k]}](https://s0.wp.com/latex.php?latex=%7Bg_k+%3D+%5Bg%27_k%2C+g%27%27_k%5D%7D&bg=ffffff&fg=000000&s=0&c=20201002)
for some 
. By induction hypothesis, we can find 
with 
for 
and 
. By surjectivity of the projection of 
, one can find 
with 
and 
. Taking commutators of these two elements, we obtain the claim.
, we conclude that 
. Similarly for permutations. Multiplying these together we see that 
are real analytic for every 
? 
are real analytic for every 
. If analyticity was preserved by smooth limits, this would imply that 
is real analytic, which would make 
. If 
going to infinity such that 
as 
(say) for infinitely many 
, hence 
for infinitely many 
be unit vectors in a Hilbert space 
for some unit complex numbers 
. By Cauchy-Schwarz we have 
for all 
and some 
. Then we have 
for at least 
of the pairs 
. 
with 
is at most 
correlate. In particular, if the vector 
also correlates with a separate collection 
of unit vectors, the pairs 
for which 
correlate (except of course on the diagonal 
where they must correlate).
be unit vectors in a Hilbert space for 
such that 
for all 
. Then there are at least 
pairs 
. In particular (by Cauchy-Schwarz) we have 
for all 
and 
, 
. 
, then there are many pairs 
for all 
is the orthogonal projection to the complement of 
denotes the claim that 
is positive semi-definite. By the Schur product theorem, we conclude that 
, 
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