Some Classical Results Related to Exponential Sums
Speaker: Shukun WuDate of Talk: September 30, 2022
Upstream link: UCLA Participating Analysis Seminar
1. Basic Definition and Motivation
This talk was centred around exponential sums, which are defined to be functions of the form:
Definition 1. Exponential Sums
Let \(e(x)=\exp(2\pi ix)\) for \(x\in\mathbb C\). An exponential sum is one of the form \(\sum_{n=1}^N e\left(f(n)\right)\) for \(N\) a positive integer and \(f\) a polynomial with real coefficients.
These sums show up frequently in Fourier analysis, and they’re also somewhat reminiscent of modular forms to me. Both of them involve sums of exponentials, and it’s a very natural question to ask: how big can these sums get in magnitude?
Using the triangle inequality will immediately get you a bound of \(N\), but this bound isn’t always good enough, and it’s far from sharp. A theorem of Wiles provides a modest improvement on the bound, but it’s still asymptotically linear in \(N\). Can we do any better?
2. Theorems of Mordell and Hua
The first big asymptotic improvement covered in the talk was as follows.
Theorem 2. (Mordell)
Let \(f\) be a polynomial with integer coefficients, and let \(p\) be a prime much larger than \(\deg f\) such that the none of the coefficients of \(f\) are not divisible by \(p\). Let \(S(p, f)=\sum_{n=1}^p e\left(\frac{f(n)}{p}\right)\). Then, \(\lvert S(p, f)\rvert \ll_k p^{1-\frac 1k}\), where \(k=\deg f\).
Note that \(f\) is now constrained to have integer coefficients rather than real coefficients, which is a significant restraint! Note additionally the division by \(p\), however; I like to think of these as “nice enough rational polynomials”. Importantly, we may indeed take the coefficients of \(\frac fp\) to fall in the interval \([0, 1)\), as we may add or subtract \(1\) from the coefficient without changing the value of the summand!
This result was generalised by Hua to the following:
Theorem 3. Hua's Estimate
Let \(f\) be a polynomial with integer coefficients, and let \(N\) be an integer much larger than \(\deg f\) such that none of the coefficients of \(f\) are divisible by \(N\). Then, if \(S(N, f)\) is defined as above, \(\lvert S(N, f)\rvert \ll_k N^{1-\frac 1k}\), where \(k=\deg f\).
This is now a realy asymptotic improvement on the linear bound, and it is possible to show that this bound is actually sharp is certain cases.
Hua’s estimate can be proved from Mordell’s theorem by observing (somehow) that \[S\left(pq, f(x)\right)=S\left(p,\frac{f(qx)}q\right)\cdot S\left(q, \frac{f(px)}p\right), \] where \(p\) and \(q\) are primes (but I believe this holds for arbitrary integers, subject to constraints on coprimality maybe). By cleverly applying the Chinese remainder theorem to get this rearrangement along with some induction, you get the result.
3. The Dispersion Method
The very brief sketch of getting Hua’s estimate appears to be fairly elementary: it rearranges an (albeit messy) sum, invokes the Chinese remainder theorem, and applies induction. The proof of Mordell’s theorem, however, uses something called the “dispersion method”, or “Linnick’s method”, or “Linnick’s dispersion method”.
The idea is that understanding \(S(p, f)\) for a specific choice of \(f\) can be very difficult, even if we constrain \(f\) to have nice integral coefficients! However, we noticed that coefficients of \(f\) only really matter modulo \(p\), as extra multiples of \(p\) can be added or subtracted off and whisked away by the mathematical cartels. Hence, all of the different choices of \(f\) can be identified with elements of \(\left(\mathbb Z/p\mathbb Z\right)^{\deg f+1}\) by reading off coefficients, minus the elements that have zeroes in any coordinates.
Although it doesn’t look much simpler, we can now bound \(\sum S(p, f)\), where the sum ranges over all of the different possibilities of \(f\). Using a few tricks here and there to loosely control the individual elements from below, an upper bound on this big sum produces an upper bound on the summands. One way of thinking about this is controlling the average or expected size of \(S(p, f)\), and since the “sample space” is finite, the size of each \(S(p, f)\) can’t stray too far from the average.
This technique gets its name because it “disperses” the work of bounding individual objects of interest to bounding the space of all of the objects of interest, and it turns out that looking at things from this higher perspective is very, very helpful sometimes. In fact, this technique or idea came up again just two and a half months later when I gave my own talk at the analysis seminar, and I think it’s one of the most valuable things I took away from any talk at the analysis participating seminar that quarter.