4 problems

Let $A(x)=\#\bigl(\mathcal{S}_3^{(\leq 3)}\cap[1,x]\bigr)$, where $\mathcal{S}_3^{(\leq 3)}$ is the bounded-exponent semigroup associated with the prime set $\mathcal{P}_3$. Sublog…

Weitsman's exponential-growth conjecture. Solutions in general have at most exponential growth.

Let $M$ be a Riemannian manifold with non-negative Ricci curvature, let $x_0\in M$ be a fixed point, and write $r(x)=d(x,x_0)$. Let $v$ be a real function. A harmonic function has…

Bourgain–Ruzsa–Zahl conjecture. For every positive integer $n$, there is a sufficiently large positive integer $m$ such that