Corollary 3.9 Fundamental theorem of finitely generated additive groups)
11.3.4 Eliminating the quadratic phase
We are now ready to finish the proof of Theorem 11.6. From (11.22) we see in particular that
|Eh∈Wb(h)Ex∈ZTh+h2+h3f(x)f(x)e(−(M(h+h3)+ξ2)ãx)| = ηO(1) for some boundedb(h). We now focus on the f(x) term and conceal many of the other terms using theb() notation, obtaining
|Eh∈W;x∈Zb(h)b(x+h)f(x)e(−Mhãx)| = ηO(1)
,
where we used the identitye(ξãx)=e(ξã(x+h))e(−ξãh) to eliminate the phase terms which were linear inx. Splittingxinto cosets ofW and using the triangle inequality we conclude
|Ey∈ZEh,x∈Wb(h)b(x+y+h)f(x+y)e(−Mhã(x+y))| = ηO(1)
, which we rewrite as
|Ey∈ZEh,x∈Wb(h,y)b(x+h,y)f(x+y)e(−Mhãx)| = ηO(1)
. By construction ofW, we know that Mhãx=M xãhfor allx,h∈W. We now divide into two cases depending on whetherFhas characteristic 2 or not. IfFhas odd characteristic, then we have
e(−Mhãx)=e
−1
2M(x+h)ã(x+h)
e 1
2M xãx
e 1
2Mhãh
. If we then set fy:W →Cto be the function
fy(x)= f(x+y)e 1
2M xãx
we conclude that
|Ey∈ZEh,x∈Wb(h,y)b(x+h,y)fy(x)| = ηO(1)
.
On the other hand, from Lemma 11.4 (after a linear change of variables) we see that
|Eh,x∈Wb(h,y)b(x+h,y)fy(x)| ≤ fyU2(W)
and thus by (11.9)
Ey∈Zfy1/2u2(W)= ηO(1)
. By Cauchy–Schwarz we conclude
|Ey∈Zfyu2(W)| = ηO(1)
.
Since theu3(W) norm controls theu2(W) norm, and the quadratic phasee(12M xã x) does not affect theu3(W) norm, we have
fyu2(W)≤ fyu3(W) = f(y+ ã)u3(W)= fu3(y+W)
and we obtain (11.14) as desired.
Now we argue for the case whenFhas characteristic 2, using an observation of Alex Samorodnitsky (private communication). SinceM is symmetric onW, the functionx→M xãxis in fact linear onW (here we rely on the characteristic 2 hypothesis). Thus we can writeM xãx=ξãxfor someξ ∈W. By passing to the
orthogonal complement ofξ inW if necessary we may assume thatξ =0, thus M xãx=0 for allξ ∈W. This allows us to find a transformation A:W →W such that Mhãx=Ahãx+Axãh; for instance, one can write M as a matrix with coefficients inF, use the hypothesisM xãxto show that the matrix has zero diagonal, and then take Ato be the upper triangular portion ofM. We then have
e(−Mhãx)=e(−A(x+h)ã(x+h))e(Axãx)e(Ahãh)
and the rest of the argument proceeds as before.
Remark 11.11 The fact that we have to pass from the original space Z to a subspace W of somewhat lower dimension is a defect of the argument. If one knew the polynomial Freiman–Ruzsa conjecture (Conjecture 5.34) one could set W =Z, which would lead to somewhat stronger results in applications.
We now comment briefly on extending these arguments to higherk, to obtain Szemer´edi’s theorem in general. At this time of writing the inverseU3 theorem has not been extended to higherk, even in the simple case of a vector space over a finite field. However, Proposition 11.10 has been extended successfully to general k:
Proposition 11.12 (Lack of uniformity implies density increment) [138] Let Z =ZN be a cyclic group of odd prime order, let k≥3, and and let f :Z →C have magnitude bounded by1be such thatEZ(f)=0andfUk−1(Z)≥ηfor some 0< η≤1. If N ≥exp(Ok(η−Ok(1)), then there exists a proper arithmetic progres- sion P in Z of length|P| =(Nck)for some absolute constant0<ck<1such that
Ex∈Pf(x)≥k
ηOk(1) . This leads ultimately to the bound
rk(ZN)=O
N (log logN)ck
(11.23) for allk≥3 and large N, whereck>0 depends only onk; in fact in [138] the explicit valueck=1/22k+9is attained. This is currently the best bound known for rk(ZN) for generalk≥4 and largeN. It is however likely that this can be improved toO((logNN)ck) based on analogy with thek=3 case.
The proof of Proposition 11.12 is quite lengthy and difficult. In principle, one wishes to induce onk, leveraging inverse theorems forUk−2 to obtain inverse theorems forUk−1. This was the strategy employed at the start of the proof of The- orem 11.6, using the simple inverse theorem (11.9) forU2 to create the partially defined derivativeξ(h), which one then obtains arithmetic structure on. Unfor- tunately this strategy has not yet been made to work even fork=5 and for the
model case of a vector space over a finite field, mainly because the inverse theo- rem forU3is much weaker than that forU2, in particular involving an unknown spaceW (or a Bohr setB), which will ultimately depend on a certain shift param- eter h in an unpleasant way. To prove Proposition 11.12, Gowers employed a slightly different approach, starting with the original function f and takingk−3
“derivatives” f →Thf f to reduce the Uk−1 norm to the U2 norm. Employ- ing the U2 inverse theorem, one then obtains a k−3-fold derivative function ξ(h1, . . . ,hk−3). The strategy is then to establish some multilinearity properties of this functionξ in order to execute a similar scheme to the one described above.
This requires a substantial amount of new combinatorial technology, not least of which is a multilinear version of the Balog–Szemer´edi–Gowers theorem, which cannot be established simply by applying the Balog–Szemer´edi–Gowers theorem separately in each variable (again because of the issue that the structures obtained in this way for one variable will depend on the other variables). See [138] for details.
Exercises
11.3.1 (Alex Samorodnitsky, private communication) Let f : Z →C, and let D:Z×Z →R+denote the quantityD(h, ξ) := |Thf f(ξ)|2. Establish the identity
ξ1,ξ2,ξ3,ξ4∈Z:ξ1+ξ2=ξ3+ξ4
Eh1,h2,h3,h4∈Z:h1+h2=h3+h4
4 j=1
D(hj, ξj)
=
ξ∈Z
Eh∈ZD(h, ξ)4.
(Hint: first show that Dis essentially its own Fourier transform.) This identity can be used as a substitute for the first part of the above argument.