# On generating functions in additive number theory, II: Lower-order terms and applications to PDEs

Julia BrandesScott ParsellKonstantinos PouliasBob Vaughan, and I recently upload our paper “On generating functions in additive number theory, II: lower-order terms and applications to PDEs,” to the arXiv. This is somewhat of a sequel to earlier work of Vaughan as well as a recent paper of Burak Erdogan and myself (see also this previous blog post). This work began during the Heilbronn Decoupling and Efficient Congruencing workshop hosted by Kevin Hughes and Trevor Wooley.

In what follows, we adopt Vinogradov’s notation and thus write ${a \ll b}$ when ${a = O(b)}$. Let ${t}$ and ${x}$ be real numbers. We are interested in the exponential sum

$\displaystyle S(t,x) := \sum_{1 \leq n \leq N} e(tn^3 + nx) , \ \ \ e(\theta) : = e^{2 \pi i \theta}.\ \ \ \ \ (1)$

We could replace the exponent 3 by any other integer exponent, but we choose 3 for simplicity, as well as being the case of most interest. Weyl obtained, using what is now known as Weyl differencing, estimates for (1), showing that for almost all ${t}$, and every ${\epsilon >0}$

$\displaystyle \sum_{1 \leq n \leq N} e(tn^2 + nx) \ll N^{1/2+\epsilon} , \ \ \ \sum_{1 \leq n \leq N} e(tn^3 + nx) \ll N^{3/4 + \epsilon}.\ \ \ \ \ (2)$

The quadratic estimate is best possible, as is seen from Parseval. We expect a matching estimate for the cubic exponential sum in (2), but (2) is still the best that is known today. The current project originally started in an attempt to understand better three different proofs of (2). We suppose

$\displaystyle t = a/q + \beta_t , \ \ \ |\beta_t| \leq \frac{1}{q^2}, \ \ \ (a,q) = 1,\ \ \ \ \ (3)$

and then choose ${j}$ so that

$\displaystyle |x - j/q| \leq 1/(2q).\ \ \ \ \ (4)$

By uncertainty principle heuristics, we expect ${S}$, as defined in (1), to be constant on scales

$\displaystyle |t-a/q| \leq \frac{1}{N^3} , \ \ \ |x-j/q| \leq \frac{1}{N}.$

On the other hand ${S(a/q , j/q)}$ is easy to calculate:

$\displaystyle S(a/q , j/q) =\frac{N}{q} \sum_{j=1}^q e(a/q j^3 + nj/q) + O(q).$

In the circle method, one needs precise versions of this heuristic. For instance, it is known (see Theorem 7.2 in Vaughan’s book) that

$\displaystyle S(t,x) = S(a/q , j/q)\frac{ I(\beta_t , \beta_x)}{N} + \Delta,\ \ \ \ \ (5)$

where

$\displaystyle I(\beta_t , \beta_x) : = \int_{1}^N e(\beta_t u^3 + \beta_x u)du.$

and the error term satisfies

$\displaystyle \Delta \ll q(1 + |\beta_x| N + |\beta_t |N^3),\ \ \ \ \ (6)$

Note that the estimate, (6), becomes nontrivial only if ${\beta_x N}$ and ${\beta_t N^3}$ are much less than 1 as predicted by the uncertainty principle. A proof of (6) follows relatively quickly from summation by parts. One should think of the main term as being of size roughly ${N/\sqrt{q}}$, in light of Gauss sum estimates.

Vaughan, Theorem 4.1, asserts the improvement, in the special case ${\beta_x = 0}$

$\displaystyle \Delta \ll q^{1/2 + \epsilon} (1 + N^3 |\beta_t|)^{1/2}.\ \ \ \ \ (7)$

It is worth noting that (7) allows for a proof of Weyl’s inequality, (1). Unfortunately the appearance of the ${N^3}$ (as opposed to a ${N^2}$) seems to prevent one from going beyond Weyl’s inequality (perhaps because one is using a major arc technique to prove a minor arc result). One consequence of our main result below allows for a proof of Weyl inequality in a similar manner with the presence of a nontrivial linear phase.

Brüdern and Robert improved (6) to

$\displaystyle \Delta \ll q^{2/3 + \epsilon} (1 + |\beta_x| N + |\beta_t |N^3)^{1/2}.\ \ \ \ \ (8)$

In the present paper we show that the estimate (7) still holds in the case when ${\beta_x \neq 0}$, provided one takes into account addition main terms, thus improving (7).

${\ }$ Theorem 1: Suppose ${t}$ and ${x}$ are as above, as in (3) and (4). Then

$\displaystyle S(t,x) =\sideset{}{^\dag} \sum_{\substack{d | q \\ ([[j/d]] , q/d) = 1}} S(d[[j /d]] / q , a/q) \frac{I(\beta_t, x - d [[j/d]] )}{N} + \Delta,$

where

$\displaystyle \Delta \ll q^{1/2 + \epsilon} (1 +\beta_t N^3)^{1/2}.$

Here ${[[z]]}$ is the nearest integer to ${z}$ and ${\sideset{}{^\dag}\sum}$ indicates the sum is over distinct values of ${d [[j/d]]}$. ${\spadesuit}$ ${\ }$

The key point is that there are multiple main terms in Theorem 1, given in the sum, which allows an improvement of the error term in (7)

Theorem 1 also indicates that one may be able to perform a major arc only analysis of the system of equations

$\displaystyle \sum_{j=1}^4 x_i^k - y_i^k, \ \ \ k = 1,3,$

though we do not pursue this in the current paper. We are also able to use Theorem 1 to estimate the following.

${\ }$ Theorem 2: Let ${k = 2}$ or ${k = 3}$. For a.e. ${c \in \mathbb{R}}$ and every ${\epsilon >0}$

$\displaystyle \sup_{N \geq 1} \frac{1}{N^{3/4+\epsilon}} \sup_{t} | \sum_{1 \leq n \leq N} e(t n^k + (t+c)n| \ll 1.\ \ \ \ \ (9)$

Furthermore (9) is best possible, in the sense that one cannot replace the exponent ${3/4}$ with anything smaller. ${\spadesuit}$ ${\ }$

One can interpret the condition in (9) as the supremum of ${S(t,x)}$ such that ${(t,x)}$ lies on the diagonal line (on the torus), ${u+v = -c}$. Naively one would expect square root cancellation in (9), though it turns out this is not the case. The point is that square root cancellation is a heuristic that only applies to minor arcs, while the supremum in (9) is obtained on a major arc. Thus it is natural to utilize estimates that can handle both minor and major arcs simultaneously, such as the one in Theorem 1. Combining Theorem 2 with previous joint work with Burak Erdo\u gan, we can provide bound for the fractal dimension of the real and imaginary parts of the solution to Airy’s equation,

$\displaystyle u_t + u_{xxx} = 0 , \ \ \ u(0,x) = g(x).\ \ \ \ \ (10)$

${\ }$ Corollary 1 (Fractal Dimension of Airy): Let ${u(t,x)}$ be the solution to (10) with ${g}$ a non-constant characteristic function of an interval. Then the maximum fractal dimension of the real and imaginary parts of ${u(t,x)}$ restricted to

$\displaystyle \{(t,x) \in \mathbb{T}^2 : t+x = -c\},$

is at most ${17/8}$ for a.e. ${c}$. ${\spadesuit}$ ${\ }$

The introduction of the aforementioned joint work contains a detailed motivation and history to this problem. It was also explored a bit in two undergraduate projects, and one can find some pictures they generated in a 2018 project and another in a 2019 project. We finish by noting the bound of ${17/8}$ in Corollary 1 does not match the known (and expected) lower bound of ${3/4}$.