# On discrepancy properties of $\{0,\pm1\}$ sequences arising from cyclotomic polynomials

Given $n=2^k p^r q^m$ take a $d\in\Bbb Z$ with $d\mid n$ such that each $a_i$ is in $\{0,\pm1\}$ in $$f_{d,n}(x)=\frac{x^n-1}{\Phi_d(x)}=a_0+a_1x+\cdots+a_{\deg(f_{d,n}(x))}x^{\deg(f_{d,n}(x))}.$$

What is the minimum $\gamma$ and the maximum $\tau$ at which

1. $\forall t\not\equiv t'\bmod(1+ \deg(f_{d,n}(x)))$ $\exists i\in\Bbb Z$ with $0<|i|\leq\gamma$ such that $a_{t+i}\neq a_{t'+i}$ holds

2. $\exists t\not\equiv t'\bmod(1+ \deg(f_{d,n}(x)))$ such that $\forall 0\leq i\leq\tau$ $a_{t+i}=a_{t'+i}$ holds

where $t+i,t'+i$ are modulo $1+\deg(f_{d,n}(x))$?

• The $a_t$ that you ask about in question 1 are the same as the $a_t$ that you implicitly define in question 2? Also, this relates in some way to your other question about these quotients? – Gerry Myerson Jan 15 '18 at 18:28
• @GerryMyerson yes $a_t$ are from 2. (I think I need to rephrase) and yes both are related. – T.... Jan 15 '18 at 18:31
• Maybe you should link each question to the other? – Gerry Myerson Jan 15 '18 at 23:25
• @GerryMyerson I have cross-linked posts. – T.... Jan 16 '18 at 0:47