Overview
The final project will be a 5–10 page paper. There will be a first draft due in later April and the final draft due the last week of classes. There are a few different kinds of papers. I’ll describe some options here.
Project types
- Exploring a bigger problem. On the homework and in lecture, I mentioned a number of bigger and more challenging problems. You could explore one of these as a final project. Some of these problems I have an idea how to do, and some are really open research problems. In your project, you don’t necessarily have to solve the problem you’re exploring—we can’t really control that. But you should try some things and write up what you tried in a rigorous way. There is a list of possible problems below.
- Reading further into the literature. We have mentioned a number of topics that are related to projection theory that we don’t have time to discuss in detail in class. You can read about one of them and write a survey about it. You could also start with an issue discussed in class that you found muddy, and your final project could be a survey paper explaining it better. You could work on the paper by a combination of thinking the issues through yourself and reading about them in the literature in different references. There is a list of possible reading ideas below. (Some projects might involve a combination of reading and exploring. That’s certainly fine.)
Some Possible Questions to Explore
- Contagious structures for projections. In class we used Plunnecke inequality and Ruzsa inequality to prove contagious structure for projections of \(A \times A \subset \mathbb{F}_p^2\). Are there similar results for projections of an arbitrary set \(X \subset \mathbb{F}_q^2\)? Here is a precise question. Suppose that \(| \pi_t(X) | \le K |X|^{1/2}\) for \(t= 0, \infty, t_1\), and \(t_2\). Does it follow that \(| \pi_{t_1 + t_2} (X)| \le K^C |X|^{1/2}\) for a universal constant \(C\)? (What \(C\) can you get?) Similarly for \(|\pi_{t_1 t_2}(X)|\) and \(| \pi_{-t}(X)|\). See Lecture 11. (Possible reference: Katz-Tao’s work on “sums differences”)
- Projections in algebraically independent directions. Suppose that \(D = {0, 1, \infty, t_1, …, t_r} \subset \mathbb{R}\). Let \(\pi_t(x_1, x_2) = x_1 + t x_2\). Let \(X\) be a finite subset of \(\mathbb{R}^2\). Define
\[S_D(N) = \min_{|X| = N} \max_{t \in D} | \pi_t(X)|.\]
If \(t_1, …, t_r\) are algebraically independent over \(\mathbb{Q}\), what upper and lower bounds can you prove on \(S_D(N)\) (in terms of \(N\) and \(r\))? You might want to start with \(r=1\).
- Optional question from problem set 5, related to Bombieri-Vinogradov. In problem set 5, using the large sieve, we proved the following estimate. If \(X \subset [N]\), then for \(90\%\) of \(p \in P_{N^{1/2}}\),
Inequality 1. \(\Vert (\pi_p 1_X)_h^{*2} \Vert_{L^\infty(\mathbb{Z}_p)} \lessapprox |X|.\)
This bound is sharp when \(X\) is an arithmetic progression of length \(N^\alpha\) with \(\alpha < 1/2\). But in this case, \(\Vert 1_X^{*2} \Vert_{\ell^\infty}\) is itself large. Suppose that \(X \subset [N]\) with \(|X| \sim N^{1/2}\), and suppose that \(\Vert1_X^{*2} \Vert_{L^\infty} \lessapprox 1\). For most \(p \in P_{N^{1/2}}\), can we prove a bound for \(\Vert(\pi_p 1_X)^{*2}_h \Vert_{L^\infty(\mathbb{Z}_p)}\) which improves on Inequality 1?
- Optional question from problem set 4, related to the large sieve. To pursue this direction, it would be helpful to have a little background in restriction theory in Fourier analysis. In class, we used the large sieve to prove the following estimate.
Theorem 1. If \(X \subset [N]\) and \(| \pi_p(X) | \le (0.99) p\) for every \(p \in P_{N^{1/2}}\), then \(|X| \lessapprox N^{1/2}\)
This theorem is essentially sharp when \(X\) is the set of squares. We could explore what happens if we know \(| \pi_p(X) \le (0.99) p\) for every \(p \in P_{N^\alpha}\) for some other exponent \(\alpha\), such as \(\alpha = 1/4\). Or we could explore what happens if we replace \(| \pi_p(X)| \le (0.99) p\) by a stronger bound like \(|\pi_p(X)| \le N^{1/4}\) for every \(p \in P_{N^{1/2}}\).
- Non-commutative projection theory. We have presented projection theory in the context of commutative groups. The setting is that we have a commutative group \(G\) and many homomorphisms \(\pi_j: G \rightarrow H_j\). Each homomorphism can be described by its kernel, \(K_j\). So \(\pi_j: G \rightarrow G / K_j\). Now suppose that \(G\) is a non-commutative group. Let \(K_j\) be a bunch of subgroups, and consider the maps \(\pi_j G \rightarrow G / K_j\). How much of what we discussed in class can be generalized to this setting? It might help to think in general or to pick a simple non-commutative group, such as \(SL_2(\mathbb{F}_p)\). Projection theory for general commutative groups \(G\) is also a possible project to explore.
- Something else that you think of.
Some Reading Ideas
Reading on the central limit theorem for convex bodies. See The Central Limit Problem for Convex Bodies and A Central Limit Theorem for Convex Sets.
The sum-product theorem in finite fields by Bourgain-Katz-Tao. We discussed many parts of this paper, but there is a little more to read to get the full sum-product theorem, saying that if \(A \subset \mathbb{F}_p\) with \(|A| = p^s\) and \(0 < s < 1\), then \(\max(|A+ A|, |A \cdot A|) \ge p^s + \epsilon(s)\). See A Sum-Product Estimate in Finite Fields, and Applications.
Plunnecke inequality. Reading the older proof of Plunnecke-Ruzsa and comparing it to the slick proof that we saw in class. See chapter 1 of Sumsets and Structure.
Suppose that \(\pi_p: \mathbb{Z}^d \rightarrow \mathbb{Z}_p^d\) is reduction modulo \(p\). Suppose that \(A \subset \mathbb{Z}^d\) and that \(| \pi_p(A)|\) is small for many primes \(p\). The known examples of this phenomenon have a lot of algebraic and number theoretic structure. The inverse problem for the large sieve asks whether all such examples have a lot of structure. The most interesting work on the problem was done by Miguel Walsh, and could be a good reading project. See The Inverse Sieve Problem in High Dimensions.
The Bombieri-Vinogradov theorem. We discussed some of the ideas in class. Try to fill in the details and perhaps compare your work with proofs in the literature. For a source from the literature, see the book by Iwaniec Kowalski or
The Large Sieve and the Bombieri-Vinogradov Theorem.
(If you can’t read that whole thing, google “Tao Bombieri Vinogradov”.)
