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 $\Bbb Z^s$?




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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 $\min\_{\substack{
k\in\Bbb Z\\ta\_i-kp\leq0}}(-ta\_i + kp)$?



Essential clarifications on application of pigeonhole principle
First, we prove the Lemma 4. Considering the linked paper (that was in the previous version of this question), $p$ must be a prime number. 


We apply the pigeon-hole principle in the following way: 


Take $T\_i'$ to satisfy $T\_i\leq T\_i'$ and $p/T\_i' \in \mathbb{N}$. This is possible without changing $T\_i$ too much (still satisfying $T\_i\asymp T\_i'$). Subdivide the interval $[0,p]$ into $p/T\_i'$ equal pieces. Considering the subdivision in each component $i$, we have a subdivision of the $s$-cube $[0,p]^s$ into $N=\prod (p/T\_i')$ equal boxes. Now given $\prod T\_i > p^{s-1}$ gives $\prod T\_i'> p^{s-1}$. Then the number of boxes inside $[0,p]^s$ with the subdivision is $<p$. Consider the following for $t=0, 1, \ldots, p-1$:
$$
(a\_1, \ldots, a\_s) t \ \mathrm{mod} \ p,
$$
**Here comes the key argument in the pigeon-hole principle.**


Since the number of boxes $N$ is $<p$, there must be two distinct $t\_0, t\_1 \in \{0, 1, \ldots p-1\}$ such that $(a\_1, \ldots, a\_s) t\_0 \ \mathrm{mod} \ p$ and $(a\_1, \ldots , a\_s) t\_1 \ \mathrm{mod} \ p$ lie in the same box. Take $t=|t\_0-t\_1|$, then $0<t<p$ so that $(t,p)=1$ and for each $i=1, \ldots, s$, 
$$
a\_i|t\_0-t\_1| \ \mathrm{mod} \ p \textrm{ is in the interval } [-T\_i' , T\_i'].
$$


Now for your questions, this argument does not guarantee the specific location of $(a\_1, \ldots , a\_s) t$ modulo $p$ other than just a box around the origin. So, in your setting of specified box, it might not be possible. With a little care, you can make a counterexample (e. g. with $s=2$, $p=37$, $a\_1=1$, $a\_2=2$, $|I\_1|=37/6$, $|I\_2|=37/6$ where $|I|$ is the length of the interval $I$). 


For using $\min\_{k\in \mathbb{Z} \\ a\_i t - kp\geq 0} (a\_i t - kp)$ or $\min{k\in\mathbb{Z}\\ a\_i t - kp \leq 0 } (-a\_i t + kp)$, we also see that the calculation modulo $p$ does not guarantee the specific sign of $a\_i t - kp$. So, it is not possible to replace by those. 


