 Unit equation over a number field

For a number field $K$ and fixed nonzero coefficients $\alpha ,\beta \in K$, the equationhas finitely many solutions in the unit group of its ring of integers. The coefficients remain fixed; varying them gives a different finiteness problem.

Here is the logarithmic-form mechanism. The Dirichlet unit theorem writes each unit as a root of unity times a product of fixed fundamental units with integer exponents. If the unit rank is zero there are already only finitely many possibilities. Otherwise set $B$ to the maximum of $3$ and the absolute values of their exponents. The logarithmic embedding and the product formula imply that a unit with a large exponent vector is exponentially small at some archimedean place, after possibly interchanging $u$ and $v$.

At that place, $\alpha u+\beta v=1$ makes the other term exponentially close to one. A nonzero logarithm of that term is a linear form in logarithms of algebraic numbers, including a suitable multiple of $2\pi i$ when complex logarithm branches are adjusted. Its integer coefficients are $O(B)$. The Baker lower bound for a homogeneous linear form in logarithms bounds its nonzero size below by $B^{-C}$ for fixed data, contradicting an exponentially small upper bound when $B$ grows. The logarithm cannot vanish, since then the other term would be exactly one and the remaining nonzero term zero. Thus the exponent vectors are bounded, leaving finitely many solutions. A finite torsion subgroup and finitely many archimedean places allow uniform constants.