Let $X_1$, ..., $X_n$ be a list of real numbers.

Consider an integer relation equation

$A_1 X_1 + \ldots + A_n X_n = 0$

where $A_1$, ..., $A_n$ are unknown integers.

Suppose somehow we are not so much interested in exact solution of the relation as we are interested in making the $|A_1 X_1 + \ldots + A_n X_n|$ as small as possible.

Question 1: what is an upper bound (in terms of $\epsilon$) on the norm $|[A_1,\ldots,A_n]|$ for which $|A_1 X_1 + \ldots + A_n X_n| < \epsilon$ ?

Additional considerations:

0) Vector $X_1$, ..., $X_n$ is assumed to be in general position

1) it is ok to assume that $|[X_1,...,X_n]|=1$

2) I will be happy to learn an answer in the specific context of the PSLQ algorithm

3) and/or I will be happy to know the answer for $n=4$

4) if this helps...

Question 2: What would be some general method(s) to investigate Question 1?

Alex--

simultaneousDiophantine approximations. This is an entirely different animal. Even if you manage to piece together an integer relation from the simultaneous Dirichlet, its quality is going to be quite bad. $\endgroup$ – Alex Mar 31 '14 at 22:58