An important theorem in Diophantine approximation is the theorem of Liuoville:
**Liuoville Theorem** If x is a algebraic number of degree $latex n$ over the rational number then there exists a constant $latex c(x) > 0$ such that:$latex \left|x-{\frac {p}{q}}\right|>{\frac {c(x)}{q^{{n}}}}$
holds for every integer $latex p,q\in N^*$ where $latex q>0$.
This theorem explain a phenomenon, the approximation of algebraic number by rational number could not be very well. Which was generated later to **Thue–Siegel–Roth theorem**, them could be used to proof a lots of constant is not algebraic, i.e. transcendentals .
My questions is in another direction, now let us not just consider one root $latex \alpha_1$ of a integer polynomial $latex P(x)=a_mx^m+…+a_1x+a_0$ but consider all roots of it, i.e. $latex \{\alpha_1,…,\alpha_m\}$, which is based on a observation : If we define
$latex \sigma_k(P(x))=\sum_{1\leq \alpha_{i_1}<\alpha_{i_2}<…<\alpha_{i_k}\leq m}\alpha_{i_1}\alpha_{i_2}…\alpha_{i_k}$
By **Vieta theorem** we know $latex \sigma_k(n)\in \mathbb Q$ for all $latex k\in N^*$, this will lead to some restriction and in fact destroy the uniformly distribution of $latex (\alpha_1,…,\alpha_m)\in [0,1]^m$. In fact the most important one is the determination of Vandermon Determinant:
$latex V(P(x))=\Pi_{1\leq \alpha_i<\alpha_j\leq m}(\alpha_i-\alpha_j)$.
We know $latex \Pi_{1\leq \alpha_i<\alpha_j\leq m}(\alpha_i-\alpha_j)\in \mathbb Q$ so when $latex \Pi_{1\leq \alpha_i<\alpha_j\leq m}(\alpha_i-\alpha_j)\neq 0$ we could use this to proof a nontrivial estimate for $latex \sum_{1\leq k\leq m}||\alpha_kn||_{\mathbb R/\mathbb Z}$.
$latex \sum_{1\leq k\leq m}||\alpha_kn||_{\mathbb R/\mathbb Z}= O(\frac{1}{n^{\frac{1}{m-1}}}).$
by combine the A-G inequality and $latex \Pi_{1\leq \alpha_i<\alpha_j\leq n}(\alpha_i-\alpha_j)=\lambda\neq 0$.While by continue fractional expansion we only know a trivial estimate of type $latex \sum_{1\leq k\leq m}||\alpha_kn||_{\mathbb R/\mathbb Z}= O(\frac{1}{n})$.
my question is the following:
Is there still have a nontrivial estimate for $latex \sum_{1\leq k\leq m}||\alpha_kn||_{\mathbb R/\mathbb Z}$ (which could be slight weaker), if we don’t have the whole power of **Vieta theorem**? more precisely:
**problem 1**
if we have $latex \sigma_k((\alpha_1,…,\alpha_m))=\lambda_k\in \mathbb Q$ for all $latex k\in \{1,2,…,m’\}$ where $latex m'<m$, is there still some nontrivial estimate of,
$latex \sum_{1\leq k\leq m}||\alpha_kn||_{\mathbb R/\mathbb Z}$
hold for all $latex n\in N^*$?
One reason to consider this could be true is that although $latex \{\alpha_1,…,\alpha_m\}$ is not roots of a integer polynomial but we could image in some suitable metric space $latex X$ the gromov-hausdorff distance of tuple $latex (\alpha_1,…,\alpha_m)$ and a tuple come form roots of integer polynomial is small . And it seems reasonable to image this type of asymptotic quality is continue with the G-H distance on $latex X$.
Another problem is what happen when $latex V((\alpha_1,…,\alpha_m))=\Pi_{1\leq i<j\leq n}(\alpha_i-\alpha_j)=0$. More precisely,
**problem 2**
What happen when $latex V((\alpha_1,…,\alpha_m))=\Pi_{1\leq i<j\leq n}(\alpha_i-\alpha_j)=0$ , is this result,
$latex \sum_{1\leq k\leq m}||\alpha_kn||_{\mathbb R/\mathbb Z}= O(\frac{1}{n^{\frac{1}{m-1}}}).$
still true?
Let us go a litter further, if these two problem both have a satisfied answer, what is the higher dimensional case?
**problem 3**
Given $latex m\in \mathbb N^*$. If tuple $latex (y_1,…,y_k)$ is very closed to the zero set of a variety in $latex \mathbb Z[x_1,…,x_m]$ in $latex \mathbb (Z^{m})^k$ in the sense a lots of symmetric sum of $latex y_1,…,y_k$ belong to $latex \mathbb Q^m$, will this lead to some good estimate for
$latex \sum_{1\leq s\leq k}||y_sn||_{\mathbb R^m/\mathbb Z^m}?$
I think these type of result should be investigated very well, Iappreciate to any pointer with useful comments and answer, both on given some strategy to solve these problems or given some reference about these problems.