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 $|A_1 X_1 + \ldots + A_n X_n| < \epsilon$ ?
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?