2 problems

Let $(\alpha,\beta)$ be a pair of real numbers, and let $\epsilon>0$. Consider the system … A pair is called multiplicatively singular if, for every $\epsilon>0$, this system has a…

Let $p$ be a prime and let $b1$ be an irrational real number. Let $\lvert q\rvert_p=p^{-v_p(q)}$ denote the $p$-adic absolute value of the integer $q$, where…