In this blog post, I would like to specialise the arguments of Bourgain, Demeter, and Guth from the previous post to the two-dimensional case of the Vinogradov main conjecture, namely
This particular case of the main conjecture has a classical proof using some elementary number theory. Indeed, the left-hand side can be viewed as the number of solutions to the system of equations
with . These two equations can combine (using the algebraic identity applied to ) to imply the further equation
which, when combined with the divisor bound, shows that each is associated to choices of excluding diagonal cases when two of the collide, and this easily yields Theorem 1. However, the Bourgain-Demeter-Guth argument (which, in the two dimensional case, is essentially contained in a previous paper of Bourgain and Demeter) does not require the divisor bound, and extends for instance to the the more general case where ranges in a -separated set of reals between to .
In this special case, the Bourgain-Demeter argument simplifies, as the lower dimensional inductive hypothesis becomes a simple almost orthogonality claim, and the multilinear Kakeya estimate needed is also easy (collapsing to just Fubini’s theorem). Also one can work entirely in the context of the Vinogradov main conjecture, and not turn to the increased generality of decoupling inequalities (though this additional generality is convenient in higher dimensions). As such, I am presenting this special case as an introduction to the Bourgain-Demeter-Guth machinery.
We now give the specialisation of the Bourgain-Demeter argument to Theorem 1. It will suffice to establish the bound
for all , (where we keep fixed and send to infinity), as the bound then follows by combining the above bound with the trivial bound . Accordingly, for any and , we let denote the claim that
as . Clearly, for any fixed , holds for some large , and it will suffice to establish
Proposition 2 Let , and let be such that holds. Then there exists (depending continuously on ) such that holds.
Indeed, this proposition shows that for , the infimum of the for which holds is zero.
We prove the proposition below the fold, using a simplified form of the methods discussed in the previous blog post. To simplify the exposition we will be a bit cavalier with the uncertainty principle, for instance by essentially ignoring the tails of rapidly decreasing functions.
Henceforth we fix and , and assume that holds. For any interval , let denote the exponential sum
whenever are disjoint intervals in that are separated by . Indeed, suppose the bilinear estimate (4) held for all . If we define the quantity
then by decomposing into intervals of length about , with a moderately large natural number, we can use the triangle inequality to bound
By (4), the contribution of those with is . On the other hand, by Hölder’s inequality and affine rescaling, the contribution of the near-diagonal with is . This gives the inequality
Let be as in (4). For any fixed and , we let denote the best constant for which one has the bound
as , where for , ranges over a partition of into intervals of length , and
is the local norm of near , where is the rectangle
(Actually, to make the argument below work rigorously we have to replace the indicator by a smoothed out variant , but to simplify the exposition we shall simply ignore this technical issue.) The function has Fourier support in the rectangle , and so by uncertainty principle heuristics one morally has (ignoring the technical issue alluded to above) a pointwise bound of the form
It remains to establish (6). This will follow from the following claims.
- (i) (Hölder) The functions and are convex non-increasing in .
- (ii) (Rescaled induction hypothesis) We have .
- (iii) ( decoupling) We have .
- (iv) (Bilinear Kakeya) We have .
Let us now see why this proposition implies (6) for all . From the proposition we have
for sufficiently small . On the other hand, from (ii) we have . Interpolating using (i) and the hypothesis , we have
for sufficiently small and for some depending only on . Applying (iv) followed by (i) we conclude that (6) holds with replaced by . Iterating this, we can obtain (6) for arbitrarily large , as required.
The claim (i) is an easy application of Hölder’s inequality; we now turn to the more interesting claims (ii), (iii), (iv).
— 1. Rescaled induction hypothesis —
To prove (ii), we need to show
where ranges over a partition of into intervals of length , and similarly for . By Hölder’s inequality it suffices to show that
for . Since , we can use Minkowski’s inequality to conclude that
and the claim then follows from (2) (since there are intervals to sum over).
— 2. decoupling —
To prove (iii), it will suffice to show that
where the and are partitions of into intervals of length and respectively. This will follow from the pointwise estimates
for any , any and any interval of length (assuming the intervals are nicely nested in some dyadic fashion for simplicity). This expands as
where is a rectangle of dimensions roughly with sides parallel to the coordinate axes. Without the localisation to , this would be immediate from the orthogonality of the . Morally, the localisation to introduces a Fourier uncertainty by a rectangle of dimensions roughly . But the frequencies that the are Fourier supported in are essentially disjoint in even up to this uncertainty, so the global orthogonality of the should localise to the scale of the rectangle . (This can be made rigorous using suitable smoothed approximants to the indicator of , but we omit this technical detail here.)
— 3. Bilinear Kakeya —
To prove (iv), it will suffice to show that
as , where ranges over a partition of into intervals of length . By averaging, it suffices to show that
whenever is a rectangle of dimensions essentially with sides parallel to the axes. If we set , then we morally have
on , and so the estimate will follow if we can show that
(As before, to be rigorous we need to replace the localisation with a smoother weight , but we ignore this technicality here.) We now apply a logarithmic pigeonholing (conceding a factor of ) to restrict to a set in which all the means are comparable to each other, and similarly to restrict to a set where the means are comparable to each other. We can then normalise so that
Since , we have
for , so it suffices to show that
By the triangle inequality, it suffices to show that
Recall that is a rectangle of dimensions about . As each is an interval of length about , we see from the uncertainty principle that the are essentially constant along parallelograms with a horizontal side of length and a vertical height of that fit inside the rectangle in a certain orientation (depending on the location of ; the slanted side has vertical slope ). Thus the functions also exhibit similar behaviour, and can be essentially written within as
for some non-negative coefficients and some parallelograms of horizontal side and height in . The estimate (7) then takes the form
so it would suffice (since ) to show that
for any parallelograms associated to intervals from respectively. But the transversality of ensures that these parallelograms have vertical slopes that differ by , and the claim follows from simple geometry ( behaves like a parallelogram of horizontal side and height ).