Let $(x_n)_{n=1}^{\infty}$ be a sequence in $\mathbb{R}$ such that $x_n\rightarrow+\infty$ as $n\rightarrow\infty$. Write $\phi\colon\mathbb{R}\rightarrow\mathbb{R}/\mathbb{Z}$ for the projection map. Is it true that for almost all $t\in\mathbb{R}$(in the usual Lebesgue measure), the sequence $(\phi(tx_n))_{n=1}^{\infty}$ is equidistributed in $\mathbb{R}/\mathbb{Z}$(in the quotient measure)? For example, for $x_n=nx$ for any $x>0$, we know this is true from the Equidistribution Theorem. Moreover Weyl's criterion gives the equidistribution for a fixed value $t$.
More generally, write $\mathbb{A}$ for the adele ring of $\mathbb{Q}$. Then for a place $v$ of $\mathbb{Q}$, we have the composition map $\phi\colon\mathbb{Q}_v\rightarrow\mathbb{A}\rightarrow\mathbb{Q}\backslash\mathbb{A}/\widehat{\mathbb{Z}}\simeq\mathbb{R}/\mathbb{Z}$ where the first map is the canonical embedding, the second map is the projection and the last map is a homeomorphism of topological groups. Given a sequence $(x_n)_n$ in $\mathbb{Q}_v$ such that $|x_n|_v\rightarrow\infty$ as $n\rightarrow\infty$. Is it true that for almost all $t\in\mathbb{Q}_v$, the sequence $(\phi(tx_n))_n$ is equidistributed in $\mathbb{R}/\mathbb{Z}$? Again for $v=p$ a finite place and $x_n=1/p^n$, we know this is true thanks to a theorem of Ferrero and Washington (see Proposition 2 of here).
I would also like to know whether we can say something if we replace 'equidistributed' by 'dense'.
I would appreciate if someone could point me to some references or indicate whether these results hold or not. By the way, I am not sure whether this is the right place to ask such questions. If not, I will delete my post.
Added: thanks to the comments below, I realised that my questions about equidistribution have a negative answer, for example, for $v=\infty$ and $x_n=\ln(n)$ or a sequence that grows very slowly. On the other hand, one can check that using Weyl's criterion, for any strictly increasing integer sequence $(x_n)_n$, the answer to the question in the case $v=\infty$ is positive.