Open-problem statements, with their sources attached.

Suppose that $X, Y$ are two finitely-supported independent random variables taking integer values, and such that $X + Y$ is uniformly distributed on its range. Are $X$ and $Y$ themselves uniformly distributed on their ranges?

Lower bound for $c(p)$ for $1 < p \le \infty$, improving the known value at $p = 2$ or $p = \infty$.

Suppose that $A$ is an open subset of $[0, 1]^2$ with measure $\alpha$. Are there four points in $A$ determining an axis-parallel rectangle with area $\gt c \alpha^2$?

Let $K \subset \mathbb{R}^n$ be a balanced compact set (that is, $\lambda K \subseteq K$ whenever $|\lambda| \leq 1$) and suppose that the normalised Gaussian measure $\gamma_n(K) \geq 0.99$. Does $10K$ contain a compact convex set $C$ with $\gamma_n(C) \geq 0.01$?

Suppose that $A$ is a $K$-approximate group (not necessarily abelian). Is there $S \subset A$, $|S| \gg K^{-O(1)} |A|$, with $S^8 \subset A^4$?

Upper bound for $c(p)$ for $1 < p \le \infty$, improving the best-known value at $p = \infty$.

Let A ⊂ R be a set of positive measure. Does $A$ contain an affine copy of {1, 1/2, 1/4, . . . }?

What is the largest product-free set in the alternating group $A_n$?

How many rotated (about the origin) copies of the 'pyjama set' $\\{(x, y) \in \mathbb{R}^2 : \text{dist}(x, \mathbb{Z}) \leq \varepsilon\\}$ are needed to cover $\mathbb{R}^2$?

In particular, can one find a better bound than the best-known bound from [KrLe25]?

Is there a better bound than the best-known bound from [KrLe25]? This is an existential version of the main problem that does not require providing the bound explicitly.

Is $\varepsilon^{-C}$ rotations enough?

Can the Cohn-Elkies scheme be used to prove the optimal bound for circle-packings in 2 dimensions?

Let $A$ be a set of $n$ positive integers. Does $A$ contain a sum-free set of size at least $\frac n 3 + Ω(n)$, where $Ω(n) → ∞$ as $n → ∞$?

Let $G$ be an abelian group of size $N$, and suppose that $A \subset G$ has density $\alpha$. Are there at least $\alpha^{15} N^{10}$ tuples $(x_1, \dots, x_5, y_1, \dots, y_5) \in G^{10}$ such that $x_i + y_j \in A$ whenever $j \in \{i, i+1, i+2\}$?

Note: We interpret indices modulo 5.

Is $W(k, r)$ a polynomial in $r$, for fixed $k$?

We formulate this as asking if $W(k, r)$ has polynomial growth in $r$. We know it is not the case for $k = 3$ [Gr21, p.3].

It remains an interesting open problem to actually write down a colouring showing (say) $W(3, r) \ge 2r^2$ for some $r$. [Gr24]

$W(3, 20) \ge 389$ from [AKS14, Table 2].

$W(3, 21) \ge 416$ from [AKS14, Table 2].

$W(3, 22) \ge 464$ from [AKS14, Table 2].

$W(3, 23) \ge 516$ from [AKS14, Table 2].