# On discrepancy of integer sequences related to Erdos-Turan-Koksma

Assume we have a sequence $a_1,\dots,a_k\in\Bbb N$ with each $a_i\approx n$ where $n$ is some integer.

Suppose there exists an $\eta\in(0,1)$ such that for every $d_1,\dots,d_k\in\Bbb Z$ with $$a_1d_1+\dots+a_kd_k=0$$ there is a $d_i$ with $|d_i|>n^\eta$ then is it true that for every prime $p\approx n^{\ell'+\eta}$ where $1\leq\ell'$ holds there is an $m\in\Bbb N$ with $n^{\eta+\ell'-1}<m<p-n^{\eta+\ell'-1}$ such that each of $$ma_i\bmod p$$ lie in $[0,n^{\ell'}]$?

Is there a way to get such a result from discrepancy theory and Erdos-Turan-Koksma result?

• What is the discrepancy of an integer sequence? What are the $c_i$? – Goldstern Feb 15 '17 at 10:04
• Something is wrong in the statement of the question. Since $\ell' \geq 1$ the numbers $m a_i \mod p$ trivially are in $[0,p] \subset [0,p^{\ell'}]$. – Kurisuto Asutora Feb 27 '17 at 11:43
• @KurisutoAsutora corrected. – user94040 Feb 28 '17 at 5:26
• I don't think that discrepancy theory and Erdös-Turan inequality are the right thing to try here. To me, this looks more like (simultaneous) Diophantine approximation. Maybe try to find a sophisticated application of the pigeon hole principle? – Kurisuto Asutora Feb 28 '17 at 8:20
• @KurisutoAsutora thank you for your feedback. Have you seen similar problem before? any thoughts? any references? – user94040 Mar 1 '17 at 1:39