5 problems

Pilatte's conjecture. These sums are bounded uniformly for all $x>0$ and all $\alpha\in E$ if and only if $f$ is eventually periodic. The conjecture was communicated to the authors…

Let $a>2$ be even and let $b$ be sufficiently large. Consider an Ulam set with generators $v_1$ and $v_2$ whose associated lattice is generated by $(-a,b)$. Regularity conjecture.…

Sequential-join period conjecture. If $\mathcal{G}_s(n+_{\ell}m)$ eventually has period $p$, then $\mathcal{G}_s(n\rhd m)$ also eventually has period $p$.

Selective-sum period conjecture. If $\mathcal{G}_s(n+_{\ell}m)$ eventually has period $p$, then $\mathcal{G}_s(n\triangledown m)$ also eventually has period $p$.

Disjunctive-sum periodicity conjecture. For every $m$, there exists $N$ such that