Unsolved Problems

Suppose that $A \subset [0, 1]$ is open and has measure greater than $1/3$. Is there necessarily a solution to $xy = z$ with $x, y, z \in A$?...

Let $A \subset \mathbb{R}$ be a set of positive measure. Does $A$ contain an affine copy of $\{1, \frac{1}{2}, \frac{1}{4}, \dots\}$?...

Let $A$ be a set of $n$ integers. Is there some $\theta$ such that $\sum_{a \in A} \cos(a\theta) \leq -c\sqrt{n}$?...

Let $A \subset \mathbb{Z}$ be a set of size $n$. For how many $\theta \in \mathbb{R}/\mathbb{Z}$ must we have $\sum_{a \in A} \cos(a\theta) = 0$?...