# Obtaining precise values from good approximation

Problem:

We would like to calculate $S=\sum_{i=1}^{k} c_i x_i$, where $k$ is a constant, $x_i$ are some fixed algebraic numbers, $c_i=\frac{p_i}{q_i}$ are rational numbers such that integers $p_i$ and $q_i$ are coprime, and their absolute value is bounded by $C^n$, where $C$ is a constant depending only on $x_i$'s and $n$ is some parameter.

Operations Allowed:

We are given the $x_i$'s, but we do not know $c_i$. We could calculate $S'$ which is $\epsilon^{n^t}$ away from the true value $S$, where $0<\epsilon<1$ is a constant, and $t$ is some fixed constant no less than 2.

The Question:

If we could show that, for all rational $c_i$, there exists only one $c_1' \ldots c_k'$, such that $\sum_{i=1}^{k} c_i' x_i' \in (S'-\epsilon^{n^t}, S'+\epsilon^{n^t})$, then obviously $c_i'=c_i$ and we could thus know $S$. Is this true and how could it be proved? Or maybe, any other approaches to calculate this $S$ from approximation?

Thank you.

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I am thinking about using the Dirichlet's approximation theorem, but I guess I need something stronger: Dirichlet's approximation says that we could approximate an irrational number with rational numbers arbitrarily well, while in the current setting, I guess we need to show that given a bound on error, there exists certain not so large denominator that approximates it well. Is it possible to show this? –  Sangxia Huang Mar 15 '10 at 2:18
We currently have a solution when $$x_i$$'s are algebraic numbers. The solution is based on some observations related to conjugates. Writing about details now. –  Sangxia Huang Mar 19 '10 at 3:10