# Discrepancy bound of integer tensor product sequence?

Here discrepancy is from $$(2.4)$$ in https://www.ricam.oeaw.ac.at/files/people/siambook_nied.pdf given by 'The (extreme) 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 $${\mathcal I}=[0,1)$$). '

Given a vector $$(v_1,\dots,v_n)\in\mathbb Z^n$$ if $$D$$ is the discrepancy of the fractional parts $$\big(\{\frac{mv_1}p\},\frac{mv_2}p\},\dots,\{\frac{mv_n}p\}\big)$$ where $$p$$ is a prime and $$m\in\{1,\dots,p\}$$ then we know that we can find an $$m$$ such that $$\big(\{\frac{mv_1}p\},\frac{mv_2}p\},\dots,\{\frac{mv_n}p\}\big)\in\mathcal I_1\times\dots\times\mathcal I_n$$ where intervals $$\mathcal I_i\subseteq(0,1)$$ satisfy condition $$\prod_{i=1}^n|\mathcal I_i|\geq D$$ holds and in particular $$|\mathcal I_1|=\dots=|\mathcal I_n|=D^{1/n}+\epsilon$$ is possible at any $$\epsilon>0$$.

If $$(v_1,\dots,v_n)=(a_1,b_1)\otimes(a_2,b_2)\otimes\dots\otimes(a_t,b_t)$$ (note $$n=2^t$$) 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 holds with $$p^{1/n}+1 then is there an $$m\in\mathbb Z$$ such that $$|\mathcal I_1|=\dots=|\mathcal I_n|=D^{1/t}+\epsilon$$ is possible at any $$\epsilon>0$$ (even though $$n=2^t$$ we only have $$t$$ constituent vectors and each coordinate has only $$t$$ degrees of freedom for tensor product sequence)?

Note discrepancy of tensor product sequence is at most $$p^{-t/n}$$ (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 $$103$$) and fact that tightness of Bombieri-Vaaler bound for this tensor sequence gives $$\max(1,|h_i|)$$ roughly $$p^{-t/n(n-1)}$$.

If $$|\mathcal I_1|=\dots=|\mathcal I_n|=p^{-1/n}+\epsilon$$ is possible then we can meet Dirichlet pigeonhole bound in Difference between Dirichlet Pigeonhole and Exponential sums bound in particular situation?.