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.