Given four positive integers $n,$ $m,$ $l$ and $k \geq 2.$ I want to find a lower bound for this expression $$|\sqrt[k]{n}+\sqrt[k]{m}-\sqrt[k]{l}|$$ in terms of these integers. Many thanks
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$\begingroup$ It could be zero. If it is not zero, then giving a lower bound is a difficult diophantine approximation problem because even the rational approximations to $\sqrt[k]{n}$ are difficult (which is the case $\sqrt[k]{p^k n}+0-\sqrt[k]{q^k}$). I am no expert however, and do not know the status of these problems. $\endgroup$– Boris BukhJul 31, 2015 at 16:52
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$\begingroup$ I actually want to find a lower bound in the case of non zero. $\endgroup$– Khadija MbarkiJul 31, 2015 at 16:55
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3$\begingroup$ Consider $k^3$ conjugates of this algebraic number, which correspond to different values of roots. Their product is integer, and does vanish only in some trivial situations. If it does not vanish, it is at least 1 by absolute value, this gives some explicit bound. $\endgroup$– Fedor PetrovJul 31, 2015 at 17:09
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3$\begingroup$ @FedorPetrov This is known as Liouville's bound. It has been improved: non-effectively, there is Roth's theorem, whereas effectively there is work of Baker and later Feld'man. However, for algebaic numbers of this special form one is likely to do better than Baker-Feldman using Pade's approximations (for example, there is work of Bombieri on approximations to $\sqrt[k]{n}$). However, I do not know what the state of the art in these questions is. $\endgroup$– Boris BukhJul 31, 2015 at 17:20
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$\begingroup$ @BorisBukh: Usually for Baker type methods to work, one fixes some algebraic irrational, and then considers approximations of suitably large height. I don't immediately see how that leads to anything better than Fedor Petrov's comment above to the problem at hand (in the general situation where $n$, $m$ and $l$ may all be large). $\endgroup$– LuciaJul 31, 2015 at 22:43
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