# Determing a number which is generated by a finite set of irrational numbers from an approximation

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?

• If the independence is only pairwise, consider $z_1=\pi+1$, $z_1=\pi+2$, $z_1=\pi+3$. Then $2\pi+4= z_1+z_3=2z_2$ Jul 29, 2021 at 17:09
• 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 $n=2$ Jul 29, 2021 at 17:12
• 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).
– lulu
Jul 29, 2021 at 17:18
• 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). Jul 29, 2021 at 17:23