# Two sequence discrepancy and smallest boxes?

Take $$p$$ to be a prime and let $$a_1,\dots,a_n\in\mathbb Z$$ be some set of integers such that discrepancy of the set of fractional parts $$\{\frac{ma_1}p,\dots,\frac{ma_n}p\}$$ with $$m\in\{1,\dots,p-1\}$$ being $$\mathcal D$$.

Then we know there is an integer $$t\in\{1,\dots,p\}$$ such that the vectors $$\{\frac{ta_1}p,\dots,\frac{ta_n}p\}$$ are in any box of size $$c\mathcal D$$ for some constant $$c>0$$.

This means for any box of size $$((c\mathcal D)^{1/n}p)^n$$ in $$[-p/2,p/2]^n$$ there is an integer $$t$$ such that $$t(a_1,\dots,a_n)\bmod p$$ lies in that box. For example there is a $$t$$ such that $$t(a_1,\dots,a_n)\bmod p$$ is in $$[2(cD)^{1/n}p,3(cD)^{1/n}p]\times [0,(cD)^{1/n}p]^{n-1}$$.

1. Is it possible to reduce the box size from $$((c\mathcal D)^{1/n}p)^n$$ to something smaller?

Take $$p$$ to be a prime and let $$a_1,\dots,a_n\in\mathbb Z$$ be some set of integers such that discrepancy of the set of fractional parts $$\{\frac{ma_1}p,\dots,\frac{ma_n}p\}$$ with $$m\in\{1,\dots,p-1\}$$ being $$\mathcal D_a$$ and let $$b_1,\dots,b_n\in\mathbb Z$$ be some set of integers such that discrepancy of the set of fractional parts $$\{\frac{mb_1}p,\dots,\frac{mb_n}p\}$$ with $$m\in\{1,\dots,p-1\}$$ being $$\mathcal D_b$$.

1. What is the smallest size such that given any such box we have $$t_a,t_b\in\{1,\dots,p-1\}$$ such that $$(t_a(a_1,\dots,a_n)+t_b(b_1,\dots,b_n))\bmod p$$ lies in that box?