# Problems

Here are a couple of problems that I am interested in and are not well-known.

Problem 1 ($100): Prove that ${|A+A| \gtrsim |A|^2 d^+(A)^{-1}}$. See this paper of mine for definitions and context. As a warm up, one may wish to prove ${|A+A| \gtrsim |A|^{5/3} d^+(A)^{-2/3}}$ for$25.

Problem 2 ($50): Prove that ${|A+LA| \geq 4|A| - O(\sqrt{|A|})}$, where ${A \subset \mathbb{Z}^2}$ is finite and ${L}$ is rotation by 90 degrees. For a warm–up,$25 for a proof or disproof that ${|A+LA| \geq 4 |A| - o(|A|)}$. See this paper of mine or problem 5 of Boris Bukh’s website. Update: The warm-up was recently solved by Akshat Mudgal in the affirmative.