Weak approximation for rational places

Reorder the places so that $v_1,\dots ,v_s$ are the finite places $p_1,\dots ,p_s$ and, if the archimedean place occurs, it is $v_r=\mathrm{\infty }$. Choose integers $N_i\ge 1$ with $p_i^{-N_i}<{\epsilon }_i$ for $1\le i\le s$. Let $D_0$ be a common positive denominator of the finite targets $a_1,\dots ,a_s$, and replace it by $D:=T{D}_0$, where $T$ is a large integer coprime to every $p_i$; this keeps every $D{a}_i$ integral and lets us later make $M\mathrm{/}D$ as small as we wish.

Put $M:={\prod }_{i=1}^s{p}_i^{N_i+v_{p_i}(D)}\mathrm{.}$ By [L2], there is an integer $y$ such that $y\equiv D{a}_i(\mathrm{m}\mathrm{o}\mathrm{d}{p}_i^{N_i+v_{p_i}(D)})(1\le i\le s)\mathrm{.}$ Then for $x_0:=y\mathrm{/}D$ one has $v_{p_i}(x_0-a_i)\ge N_i$, hence $\mathrm{\mid }{x}_0-a_i{\mathrm{\mid }}_{p_i}\le p_i^{-N_i}<{\epsilon }_i$ for every finite place in the list.

If $\mathrm{\infty }$ is not among the chosen places, then $x:=x_0$ works. Otherwise every number of the form $x_k:=x_0+kM\mathrm{/}D$ has the same finite-place congruence conditions as $x_0$, because $v_{p_i}(M\mathrm{/}D)=N_i(1\le i\le s)\mathrm{.}$ By taking $T$ in step 1.1 so large that $M\mathrm{/}D<{\epsilon }_r$, the arithmetic progression $x_0+\mathbb{Z}(M\mathrm{/}D)$ has mesh smaller than ${\epsilon }_r$, so some integer $k$ satisfies $\mathrm{\mid }{x}_k-a_r{\mathrm{\mid }}_{\mathrm{\infty }}<{\epsilon }_r$.

The chosen $x$ satisfies all requested inequalities. The density formulation is the same statement with the local targets first approximated by rational elements in each factor.