Open-problem statements, with their sources attached.

Determine the asymptotic equivalence class (theta) of m(N, k).

Determine an upper bound (big O) for m(N, k).

Determine a strict upper bound (little o) for m(N, k).

Can we improve the lower bound?

Can we improve the best upper bound?

If $A \subset \mathbb{Z}/p\mathbb{Z}$ is random, $|A| = \sqrt{p}$, can we almost surely cover $\mathbb{Z}/p\mathbb{Z}$ with $100\sqrt{p}$ translates of $A$? [Gr24]

"I do not know how to answer this even with 100 replaced by 1.01." [Gr24]"

Similar questions are interesting with $\sqrt{p}$ replaced by $p^\theta$ for any $\theta \le 1/2$. [Gr24]

NOTE: using $C p^\theta$ translates as stated makes the conjecture trivially false by the pigeonhole principle. Indeed for a set of size $p^\theta$, we cover at most $C p^{2\theta}$ elements, which is strictly less than $p$ for $\theta < 1/2$. We interpret the question as asking whether $O(p^{1-\theta})$ translates suffice. This generalizes the main conjecture where $\sqrt{p} = p^{1-1/2}$.

Does $f(r) \to \infty$? [Gr24]

The possibility that f(r) = 1 for all r has not been ruled out [Gr24]

It is not known whether f(2) = 1 [Gr24]

Does $\tilde{f}(r) \to \infty$? [Gr24]

Does $f_{\text{all}}(r) \to \infty$? [Gr24]

Which finite groups have the smallest biggest product-free sets?

We formalise this as: determine the supremum of exponents $\alpha$ such that every nontrivial finite group of order $n$ contains a product-free set of size $\geq c n^{\alpha}$ for some absolute constant $c > 0$. (The trivial group is excluded since its only product-free subset is empty.) Kedlaya [Ke97] showed that $\alpha = 11/14$ is admissible, and Green suggests this exponent may well be sharp; the candidate extremal family is the Ree groups ${}^2G_2(q)$, $q = 3^{2m+1}$.

A good model problem would be to determine the largest product-free subsets of $\mathrm{SL}_2(\mathbb{F}_p)$.

Let $A \subset \mathbb{F}_2^n$ be a set of density $\alpha > 0$. Does $10A$ contain a coset of some subspace of dimension at least $n - O(\log(1/\alpha))$?

More precisely: does there exist an absolute constant $C > 0$ such that for all $n \geq 1$ and all nonempty $A \subseteq \mathbb{F}_2^n$ with density $\alpha > 0$, the sumset $10A$ contains a coset of some subspace of dimension at least $n - C \log_2(1/\alpha)$?

The sumset $10A$ is defined as $\{a_1 + a_2 + \cdots + a_{10} : a_i \in A\}$, using the pointwise scalar multiplication notation 10 • A where • denotes the iterated addition of a set.

Note: We model $\mathbb{F}_2^n$ as Fin n → ZMod 2, which is an $n$-dimensional vector space over $\mathbb{F}_2$.

Suppose that $A \subset \mathbb{F}_2^n$ is a set of density $\alpha$. What is the largest size of coset guaranteed to be contained in $2A$?

We phrase this by asking for the exact function $F(\alpha, n)$ giving the maximum dimension of a guaranteed coset.

Suppose that $A \subset \mathbb{F}_2^n$ has density $\alpha > 1/2 - C/\sqrt{n}$. Does $A + A$ contain a subspace of co-dimension $O_C(1)$? [Sa11, Question 5.1]

Suppose that $A \subset \mathbb{F}_2^n$ is a set with an additive complement of size $K$. Does $2A$ contain a coset of codimension $O_K(1)$?

Could $2A$ even contain a coset of codimension $O(\log K)$?