2 problems

Szekély's conjecture. The set of distances between pairs of points of $A$ contains all sufficiently large distances; equivalently, there exists $T>0$ such that every $t\geqslant T$…

Let $A\subset\mathbb{N}$ have positive upper Banach density, and let $k\in\mathbb{N}$. For a finite subset $F$ of $B$, write its sum as $\sum_{n\in F}n$. Higher-order shifted finit…