Open-problem statements, with their sources attached.

It is not known whether, if $G$ is an abelian group of size $n$, there always exists a Sidon subset of $G$ of size $0.01\sqrt{n}$ [Gr24].

Another very nice old problem is whether there is a Sidon subset of $\{0, 1\}^n$ of size $N^{0.51}$, where $N = 2^n$ [Gr24].

Let $p$ be a prime and let $A \subset \mathbb{Z}/p\mathbb{Z}$ be a set of size $\lfloor \sqrt{p} \rfloor$. Is there a dilate of $A$ containing a gap of length $100\sqrt{p}$?

Even what happens in the regime $\omega(p) \sim 10 \log p$ is unclear [Gr24].

Are there infinitely many $q$ for which there is a set $A \subset \mathbb{Z}/q\mathbb{Z}$, $|A| = (\sqrt{2} + o(1))q^{1/2}$, with $A + A = \mathbb{Z}/q\mathbb{Z}$? [Gr24]

Do the following exist, for arbitrarily large $n$? An abelian group $H$ with $|H| = n^{2+o(1)}$, together with subsets $A_1, ..., A_n, B_1, ..., B_n$ satisfying $|A_i||B_i| \ge n^{2-o(1)}$ and $|A_i + B_i| = |A_i||B_i|$, such that the sets $A_i + B_i$ are disjoint from the sets $A_j + B_k$ ($j \neq k$)?

NOTE: according to [CKS05, 4.1], the conditions should be $A_i + B_j$ disjoint from $A_j + B_k$ for $i \neq k$. See green_36.variants.cks05.

Variant using the exact simultaneous double product property from [CKS05, 4.1].

Given a natural number N, what is the smallest size of a subset of ℕ that contains, for each d = 1, …, N, an arithmetic progression of length k with common difference d.

Asymptotic version: determine the asymptotic behavior of m(N, k) as N grows. The solver should determine what function f : ℕ → ℝ eventually equals (fun N ↦ (m N k : ℝ)).

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]