Open-problem statements, with their sources attached.

We conjecture that the best-known upper bound can be improved.

It seems very likely that we must have $y \ll x^{1+o(1)}$ [Gr24].

Suppose that a large sieve process leaves a set of quadratic size. Is that set quadratic?

The following very particular instance is probably the simplest [Gr24]: Suppose that $A \subset \mathbb{N}$ is a set with the property that $|A \pmod p| \leqslant \frac{1}{2}(p + 1)$ for all sufficiently large $p$. Is it true that either $|A \cap [X]| \ll X^{1/2} / \log^{100} X$, or $A$ is contained in the image of $\mathbb{Z}$ under a quadratic map $\phi : \mathbb{Q} \to \mathbb{Q}$?

Is there an absolute constant $c > 0$ such that, whenever $A ⊆ \mathbb{N}$ is a set of squares with $|A| ≥ 2$, the sumset $A + A$ satisfies $|A + A| ≥ |A|^{1 + c}$?

Suppose that $A + A$ contains the first $n$ squares. Is $|A| \geq n^{1 - o(1)}$?

It is known that necessarily $|A| \geq n^{2/3 - o(1)}$, whilst in the other direction there do exist such $A$ with $|A| \ll_C n / \log^C n$ for any $C$.

Let $p$ be a large prime, and let $A$ be the set of all primes less than $p$. Is every $x \in \{1, \ldots, p-1\}$ congruent to some product $a_1 a_2$ where $a_1, a_2 \in A$?

Is there always a sum of two squares between $X - \frac{1}{10}X^{1/4}$ and $X$? We formalize this as an eventual statement for sufficiently large real $X$.

Does Ulam's sequence have positive density?