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Here discrepancy is from $(2.4)$ in https://www.ricam.oeaw.ac.at/files/people/siambook_nied.pdf given by 'The discrepancy $D_N(P) = D_N(x_l,\dots,X_N)$ of the point set $P$ of $N$ points in $\mathbb Z^s$ is defined by $D_N(P) = D_N(\mathcal J; P)$, where $\mathcal J$ is the family of all subintervals of $\mathcal I^s$ of the form $\prod_{i=1}^s[ u_í, v_i)$ where $$D_N(\mathcal B; P) = \sup_{B\in\mathcal B}\Bigg|\frac{\sum_{i=1}^nc_{ B}(x_i)}N — \lambda_s(B)\Bigg|$$ with $c_{B}(x_i)$ being indicator function while $\lambda_s(B)=\lim_{N\rightarrow\infty}\frac1N\sum_{i=1}^Nc_{B}(x_i)$ with $\mathcal B$ being a nonempty family of Lebesgue-measurable subsets of $\overline{\mathcal I}^s$ ($\overline{\mathcal I}=[0,1]$ and ${\mathcal I}=[0,1)$).

Take the vector $w=\underbrace{v\otimes\dots\otimes v}_{k\mbox{ times}}$ where $v$ is of form $(a_1,b_1)\otimes(a_2,b_2)\otimes\dots\otimes(a_t,b_t)$ (note $length(w)=2^{tk}$) where each pair $a_i,b_j$ is coprime, each pair $a_i,a_j$ is coprime and each pair $b_i,b_j$ is coprime with each $a_i,b_j\in[n,2n]$.

If discrepancy of sequence $w$ is $f(n)$ and $p\approx\frac1{f(n)^{L/2^t}}$ is a prime where $L\leq2^{tk}$ is number of distinct elements of $w$ (example if $k=2\leq t$ then $L=3^t$) then is it true that there is an $m\in\mathbb Z$ with $m\not\equiv0\bmod p$ such that the vector $$m(v_1,v_2,\dots,v_n)\equiv(r_1,\dots,r_n)\bmod p$$ as vector in $(r_1,\dots,r_n)\in\mathbb Z^n$

  1. is in box $\Big[-1-{p}\cdot{f(n)^{\frac1{2^t}}}\quad,\quad1+{p}\cdot{f(n)^{\frac1{2^t}}}\Big]^n$

  2. is independent of $w$ over $\mathbb R$

if $2\leq k<t$?

Note discrepancy bound gives such a vector in box $\Big[-1-{p}\cdot{f(n)^{1/L}}\quad,\quad1+{p}\cdot{f(n)^{1/L}}\Big]^n$ if $p\approx\frac1{f(n)^{L/2^t}}$ holds and I am looking for a smaller box since $2^k\leq L$ holds and about as small as in Difference between Dirichlet Pigeonhole and Exponential sums bound in particular situation? which in perspective would be $p^{1-1/L}$ or $p\cdot f(n)^{1/2^t}$.

Note discrepancy is obtained from $5.12$ in https://www.ricam.oeaw.ac.at/files/people/siambook_nied.pdf by using $$\rho(V,p)=\min_{h:h\cdot V = 0 \bmod p} r(h)$$$$D_N(P)\rho(V,p)=O((\log N)^n)$$ where $V=(v_1,\dots,v_n)$ and $r(h)=\prod_{i=1}^n\max(1,|h_i|)$ (page $103$).

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