Open-problem statements, with their sources attached.

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 $\mathbb{F}_2$.

Brocard's Problem* Does $n! + 1 = m^2$ have integer solutions other than $n = 4, 5, 7$?

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.

For what functions $g(N) → \infty$ is it true that $\lvert A\cap \{1,\ldots,N\}\rvert \gg \frac{N^{1/2}}{g(N)}$ implies $\limsup 1_A\ast 1_A(n)=\infty$?

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]

Can one show that $\sum_{n\leq x}g_k(n) \sim c_k x\log x$ for some constant $c_k$?

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 $O_K(1)$?

Is it true that there is a constant $c_k$ such that for almost all $n < x$ we have $g_k(n)=c_k\log x+o(\log x)$?

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

Is it true that there are only finitely many powers of $2$ which have only the digits $0$ and $1$ when written in base $3$?

Suppose $A, B ⊆ \{1, \dots, N\}$ both have size at least $N^{0.49}$. Must the sumset $A + B$ contain a composite number?

If we only allow the digits $1$ and $2$ then $2^{15}$ seems to be the largest such power of $2$.

The no-k-in-line problem: For $N \geq k$ and $k > 2$, the AllowedSetSize is $(k - 1) N$, i. e. on an $N \times N$ subset, there is a set of $(k - 1) N$ points for which no $k$ lie on a line (and not such a set of bigger size).

How many iterations of $n\mapsto\phi(n) + 1$ are needed before a prime is reached?

Green's Open Problem 72 / No-three-in-line problem*: The no-k-in-line conjecture holds for $k = 3$.

Let $c(n)$ be the minimum number of iterations of $n\mapsto\phi(n) + 1$ before a prime is reached. What is $\Theta(c(n))$?

Does the no-three-in-line problem hold when $N$ is big enough?

Let $c(n)$ be the minimum number of iterations of $n\mapsto\phi(n) + 1$ before a prime is reached. Find the simplest function $g(n)$ such that $c(n) = O(g(n))$?

Given $n$ points in the unit disc, must there be a triangle of area at most $n^{-2+o(1)}$ determined by them?

Let $c(n)$ be the minimum number of iterations of $n\mapsto\phi(n) + 1$ before a prime is reached. Find the simplest function $g(n)$ such that $c(n) = o(g(n))$?