Consider a function $f:[0,1]\rightarrow \mathbb{C}$ and points $t_0,t_1,\ldots,t_n\in[0,1]$ \begin{equation*} f(t)=\prod_{k=1}^n\frac{(e^{2\pi i t}-e^{2\pi i t_k})}{(e^{2\pi i t_0}-e^{2\pi i t_k})} \end{equation*} I would like to derive conditions on $t_0,t_1,\ldots,t_n$ under which $|f(t)|< 1$ for all $t\in[0,t_0)\cup(t_0,1]$. (note that $f(t_0)=1$ by construction).

I'm looking for conditions like

1. smallest constant $c$ such that

$min_{k,\ell}|t_k-t_\ell|\ge c/n$

suffices, (with the distance meant to be circular that is |0.9-0.1|=0.2).

or

2. more sophisticated conditions like:

$D_{n+1}(t_0,t_1,\ldots,t_n)$ needs to be small. The discrepancy of a a finite sequence of real numbers $x_1,x_2,\ldots,x_N\in[0,1]$ is defined as \begin{equation*} D_N(x_1,x_2,\ldots,x_N)=\underset{0\le\alpha<\beta\le 1}{sup}\bigg|\frac{A([\alpha,\beta);N)}{N}-(\beta-\alpha)\bigg|, \end{equation*} with $A([\alpha,\beta);N)$ denoting the number of $x_i$ such that $x_i\in[\alpha,\beta)$ (Based on section 2 of Uniform Distribution of Sequences by Kuipers and Niederreiter).