# Discrepancy related independent vector from tensor product?

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?

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 $$103$$).