Discoveries / number-theory

Sárközy's Conjecture on Sums and Products Modulo a Prime

For $A \subseteq \mathbb{F}_p$ let $A^* = (A+A) \cup (AA)$. Sárközy conjectured that for all large primes, every set of size at least $c\sqrt{p}$ has $A^* = \mathbb{F}_p$-like covering behaviour. Disproved with an explicit construction from the classical cross-ratio orbit, together with the exact extremal value.