An important theorem in Diophantine approximation is the theorem of Liouville:

Liouville Theorem If x is a algebraic number of degree $n$ over the rational numbers then there exists a constant c(x) > 0 such that: $\left|x-{\frac {p}{q}}\right|>{\frac {c(x)}{q^{{n}}}}$ holds for every integer $p,q\in N^*$ where $q>0$.

This theorem explains a phenomenon, the approximation of algebraic numbers by rational numbers could not be very good. Which was generated later to the Thue–Siegel–Roth theorem, which could be used to prove a lot of constants are not algebraic, i.e. transcendental.

My questions is in another direction; now let us not just consider one root $\alpha_1$ of an integer polynomial $P(x)=a_mx^m+...+a_1x+a_0$ but consider all roots of it, i.e. $\{\alpha_1,...,\alpha_m\}$, which is based on an observation; if we define

$$\sigma_k(P(x))=\sum_{1\leq \alpha_{i_1}<\alpha_{i_2}<...<\alpha_{i_k}\leq m}\alpha_{i_1}\alpha_{i_2}...\alpha_{i_k}$$

By the Vieta theorem we know $\sigma_k(x)\in \mathbb Q$ for all $k\in N^*$, this will lead to some restrictions and in fact destroy the uniform distribution of $(\{\alpha_1 n\},...,\{\alpha_m n\})\in [0,1]^m$. In fact, the most important one is the determination of the Vandermonde Determinant: $$V(P(m))=\Pi_{1\leq \alpha_i<\alpha_j\leq m}(\alpha_i-\alpha_j)$$.

We know $\Pi_{1\leq \alpha_i<\alpha_j\leq m}(\alpha_i-\alpha_j)\in \mathbb Q$ so when $\Pi_{1\leq \alpha_i<\alpha_j\leq m}(\alpha_i-\alpha_j)\neq 0$ we could use this to proof a nontrivial estimate for $\sum_{1\leq k\leq m}||\alpha_kn||_{\mathbb R/\mathbb Z}$. $$\sum_{1\leq k\leq m}||\alpha_kn||_{\mathbb R/\mathbb Z}\geq c{n^{\frac{-1}{m-1}}}.$$

by combining the A-G inequality and $\Pi_{1\leq \alpha_i<\alpha_j\leq n}(\alpha_i-\alpha_j)=\lambda\neq 0$. While by continual fractional expansion we only know a trivial estimate of type $\sum_{1\leq k\leq m}||\alpha_kn||_{\mathbb R/\mathbb Z}\geq c\frac{1}{n}$.

My question is the following: Is there still a estimate for $\sum_{1\leq k\leq m}||\alpha_kn||_{\mathbb R/\mathbb Z}$ (which could be slight weaker), if we don't have the whole power of Vieta theorem? More precisely:

Problem 1 If we have $\sigma_k((\alpha_1,...,\alpha_m))=\lambda_k\in \mathbb Q$ for all $k\in \{1,2,...,m'\}$ where $m'< m$, is there still some nontrivial estimate of $$\sum_{1\leq k\leq m}||\alpha_kn||_{\mathbb R/\mathbb Z}$$ hold for all $n\in N^*$?

One reason to consider this could be true is that although $\{\alpha_1,...,\alpha_m\}$ are not roots of an integer polynomial, but we could imagine in some suitable metric space $X$ the Gromov-Hausdorff distance of tuple $(\alpha_1,...,\alpha_m)$ and a tuple coming from roots of integer polynomials is small. And it seems reasonable to imagine this type of asymptotic quality is continue with the G-H distance on $X$.

Another problem is what happens when $V((\alpha_1,...,\alpha_m))=\Pi_{1\leq i<j\leq n}(\alpha_i-\alpha_j)=0$. More precisely,

Problem 2 What happens when $$V((\alpha_1,...,\alpha_m))=\Pi_{1\leq i<j\leq m}(\alpha_i-\alpha_j)=0$$. Is this result, $$\sum_{1\leq k\leq m}||\alpha_kn||_{\mathbb R/\mathbb Z}\geq c{n^{\frac{-1}{m-1}}}.$$ still true?

Let us go a litter further, if these two problems both have a satisfactory answer, what is the higher dimensional case?

Problem 3 Given $m\in \mathbb N^*$. If tuple $(y_1,...,y_k)$ is very closed to the zero set of a variety in $\mathbb Z[x_1,...,x_m]$ in $\mathbb (Z^{m})^k$ in the sense a lots of symmetric sum of $y_1,...,y_k$ belong to $\mathbb Q^m$, will this lead to some good estimate for $$\sum_{1\leq s\leq k}||y_sn||_{\mathbb R^m/\mathbb Z^m}?$$

I think these type of results should be investigated very well, I appreciate to any pointers with useful comments and answers, both on giving some strategy to solve these problems or given some reference about these problems.

  • $\begingroup$ Are those estimates right? What is $k$ in $O(\frac{1}{n^{\frac{1}{k-1}}})$? $\endgroup$ – MTyson Dec 5 '17 at 19:27
  • $\begingroup$ What do you mean by equidistribution of specific tuple of roots? it's a fixed vector! $\endgroup$ – Asaf Dec 5 '17 at 21:03
  • $\begingroup$ @MTyson, sorry, I confused the index $m$,$n$, corrected, thanks! $\endgroup$ – Hu xiyu Dec 6 '17 at 2:54
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    $\begingroup$ I'm still not sold about your reasoning with the VnD matrix. Anyhow, after you've used the AG inequality, you've transferred the question to a question regarding products of linear forms. This is traditionally dealt with Roth/Subspace type theorems, indeed by Roth's, you can get an ineffective bound of $C_{\epsilon}n^{-(1+\epsilon)/m}$. I'm pretty sure those symmetric polynomial questions can be dealt with in a similar fashion. Question 2 is false as stated, if your polynomial have repeated roots, you might have stupid counter examples like $p(x)=((x-\alpha)(x-\beta))^{2}$. $\endgroup$ – Asaf Dec 6 '17 at 3:27
  • $\begingroup$ @Asaf, thanks a lots, I will follow your advice! $\endgroup$ – Hu xiyu Dec 6 '17 at 4:39

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