9 problems

For a finite set $A$ in a commutative ring, write … The sum-product problem asks how small these two sets can be simultaneously. Erdős–Szemerédi sum-product conjecture. For…

Strong Erdős–Szemerédi conjecture. For every $c>0$ and $\varepsilon>0$, there is a threshold $n_0$ such that, if $n\geq n_0$, then for every $n$-element set of reals $\mathcal{A}$…

Let $A$ be a finite subset of a ring. For an integer $k\geq2$, let $kA$ denote the $k$-fold sum set and let $A^{(k)}$ denote the $k$-fold product set. Erdős's many sums and product…

Sum-product conjecture. In a given ring,

Sum-product conjecture for $k=10$.

Let $alpha,beta,gamma in (0,1)$ with $beta leq alpha$. Let $A,B,C subset [0,1]$ be $delta$-separated sets with $|A| leq delta^{-alpha}$, $|B|=delta^{-beta}$, and…

Let $lpha,beta,gamma in (0,1)$, and let $A,B,C subset [0,1]$ be finite sets with $|A| = delta^{-alpha}$, $|B| = delta^{-beta}$, and $|C| = delta^{-gamma}$ for some small…

Threefold product-of-sums conjecture. For every $\epsilon>0$,

Iterated product-set sum conjecture. The $k$-fold sumset of $A^{(k)}$ should satisfy