Unsolved Problems

What is the largest $y$ for which one may cover the interval $[y]$ by residue classes $a_p \pmod p$, one for each prime $p \leq x$?...

Pick $x_1, \dots, x_k \in A_n$ at random. Is it true that, almost surely as $n \to \infty$, the random walk on this set of generators and their invers...

Find bounds in the classification theorem for approximate groups....

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

Are a positive proportion of positive integers a sum of two palindromes?...

Let $d \geq 3$ be an odd integer. Give bounds on $\nu(d)$ such that if $n > \nu(d)$ the following is true: given any homogeneous polynomial $F(\mathbf...

Finding a single solution to a polynomial equation $F(x_1, \dots, x_n) = C$ can be very difficult. What conditions on $A$ ensure that the number of su...

Is every group well-approximated by finite groups?...

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