6 problems

Sum-product conjecture. For every $\delta_{1}\in(0,1)$ and every finite, non-empty set $A\subseteq\mathbb{R}$, the displayed estimate holds. This is the one-dimensional sum-product…

Let $\alpha,\beta,\delta \in (0,1)$. Assume that $A,B,E \subset [0,1]$ are compact sets with $\dim A \leq \alpha$, $\dim B \geq \beta$, and $\dim B + \dim E \geq \dim A + \delta$.…

Let $A\subset\mathbb{R}$ be a finite set. Write $A-A=\{a_1-a_2:a_1,a_2\in A\}$ and $A\cdot A=\{a_1a_2:a_1,a_2\in A\}$. The difference-product conjecture. At least one of $|A-A|$ an…

Let $A\subset\mathbb{R}$ be a finite set. Write $A+A=\{a_1+a_2:a_1,a_2\in A\}$ and $A\cdot A=\{a_1a_2:a_1,a_2\in A\}$. The sum-product conjecture. At least one of these sets should…

Let $N(f;A,A,A)$ denote the number of points of the surface $\{f=0\}$ in $A^3$. Let $G$ be a one-dimensional abelian algebraic group, and let … An irreducible surface…

Sum-product estimate conjecture. For every subset $A\txsubsetmathbb{F}_p$ and every $varepsilon>0$,