3 problems

Let $\epsilon>0$. For rational positive numbers $a_1,\dots,a_n$ and integers $b_1,\dots,b_n$, define … Assume that $b_1\log a_1+\cdots+b_n\log a_n\ne0$. Lang's conjecture. There ex…

Let $\epsilon>0$, let $a_1,\ldots,a_n$ be positive integers, and let $b_1,\ldots,b_n$ be non-zero integers such that $a_1^{b_1}\cdots a_n^{b_n}\ne 1$. Lang–Waldschmidt conjecture.…

Let $G$ and $H$ be expansions whose corresponding recurrence relations have multiplicatively independent dominant roots. Let $H_G(n)$ and $H_H(n)$ denote the relevant expansion com…