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I have the following problem.
Say I have a finite set of irrational numbers $(z_1,...,z_n)$ that are linearly independent (in the sense that I can not multiply by a rational number to get one of the $z_j$ from another). Consider then a number $z$ in the $\mathbb Z$-span with respect to this set, i.e. $$ z = \sum_j a_j z_j, $$ where $a_j \in \mathbb Z$.

In reality I obtain the number $z$ numerically and hence only to some finite digit precission (~5).
Is there any hope (uniqueness) in obtaining the coefficients $a_j$ (i.e. $z$ itself) from this approximation? If yes, what could be suitable algorithm to determine them?

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  • $\begingroup$ If the independence is only pairwise, consider $z_1=\pi+1$, $z_1=\pi+2$, $z_1=\pi+3$. Then $2\pi+4= z_1+z_3=2z_2$ $\endgroup$
    – Henry
    Jul 29, 2021 at 17:09
  • $\begingroup$ More directly related to your question, if you only have to be within $10^{-5}$ then there will be an infinite number of solutions even with $n=2$ $\endgroup$
    – Henry
    Jul 29, 2021 at 17:12
  • $\begingroup$ For instance, $47525\times \sqrt {2}-38804\times \sqrt 3\approx 0.00001$... and you can get as close to $0$ as you like (continued fractions are useful here). $\endgroup$
    – lulu
    Jul 29, 2021 at 17:18
  • $\begingroup$ Right... of course. So it is not unique for any generator set of irrational numbers? In my problem I have actually for example $(\zeta(2),\zeta(3),\log(2))$ (where $\zeta$ is the Riemann Zeta function). $\endgroup$
    – jkb1603
    Jul 29, 2021 at 17:23

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