In here Lemma $4$ using pigeonhole says:

For $T_1,\dots,T_s\in\Bbb R$ with $1\leq T_1,\dots,T_s<p$ and $\prod_{i=1}^sT_i > p^{s−1}$ and any integers $a_1,\dots,a_s$ there is an integer $t$ coprime to $p$ such that $$\min_{ k\in\Bbb Z}|ta_i − kp| \ll T_i,\quad\quad i = 1,\dots,s$$ holds.

**First of all is this even true?** Are we interpreting $(a_1,\dots,a_s)$ as a straight line in $\Bbb Z^s$?

Every integer $s$-tuple $a_1,\dots,a_s$ has $p$ mappings by $t(a_1,\dots,a_s)\bmod p$ where $t$ ranges from $\{0,1,\dots,p-1\}$. Unless we assume some uniformity in mapping I so not see how there is a $t$ such that $t(a_1,\dots,a_s)\bmod p\in[-T_1,T_1]\times[-T_2,T_2]\times\dots\times[-T_s,T_s]$ holds? If we had $p^s-p^{s-1}$ different choices of $t$ then pigeonhole works. Since $p\ll p^s-p^{s-1}$ is this some kind of randomized pigeonhole or just an argument assuming $(a_1,\dots,a_s)$ as a straight line in $\Bbb Z^s$?

Assuming we have the needed result as in Lemma $4$ then my main query is following (which satisfy pigeonhole):

**(1)** Can we replace real numbers $T_1,\dots,T_s$ by intervals $I_1=[T_0,T_1]$, $I_2=[T_1,T_2]$,$\dots$, $I_s=[T_{s-1},T_s]$ and now have following lemma?

For any real intervals $I_1=[T_0,T_1]$, $I_2=[T_1,T_2]$,$\dots$, $I_s=[T_{s-1},T_s]$ with $$0\leq T_0,T_1,\dots,T_s<p,\quad1\leq|I_1|,\dots,|I_s|<p,\quad\prod_{i=1}^s|I_i|> p^{s−1}$$ and any integers $a_1,\dots,a_s$ there is an integer $t$ coprime to $p$ such that $$\min_{ k\in\Bbb Z}|ta_i − kp|\in I_i,\quad\quad i = 1,\dots,s$$ holds.

My second query is following (which also allows pigeonhole combinatorics to work):

**(2)** Why cannot I replace $\min_{
k\in\Bbb Z}|ta_i − kp|$ by $\min_{\substack{
k\in\Bbb Z\\ta_i-kp\geq0}}(ta_i − kp)$ or $\min_{\substack{
k\in\Bbb Z\\ta_i-kp\leq0}}(-ta_i + kp)$?

(1)possible and in(2)can we replace? $\endgroup$ – user94040 Mar 5 '17 at 16:39