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 $K \subset \mathbb{R}^N$ be a balanced compact set with normalized Gaussian measure $\gamma_\infty(K) \geq 0.99$. Does $10K$ contain a compact con...

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

Is every set $\Lambda \subset \mathbb{Z}$ either a Sidon set, or a set of analyticity?...

Let $\mathcal{F}$ be all integrable functions $f : [0, 1] \to \mathbb{R}_{\geq 0}$ with $\int f = 1$. For $1 < p \leq \infty$, estimate $c_p := \inf_{...

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 $\sum_{a \in A} \cos(a\theta) = 0$?...

Describe the rough structure of sets $A \subset \mathbb{Z}$ with $|A| = n$ and $\|\hat{1}_A\|_1 \leq K \log n$....

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