Thue's inequalities and the hypergeometric method

Following a method originally due to Siegel, we establish upper bounds for the number of primitive integer solutions to inequalities of the shape $0<|F(x, y)| \leq h$,where$F(x , y) =(\alpha x + \beta y)^r -(\gamma x + \delta y)^r \in $\mathbb{Z}$[x ,y]$,$\alpha$,$\beta$,$\gamma$and$\delta$ are algebraic constants with $\alpha \delta -\beta \gamma \mathrm{\ne }0$, and $r\ge 3$ and $h$ are integers. As an important application, we pay special attention to the binomial Thue’s inequaities $\mathrm{\mid }a{x}^r-b{y}^r\mathrm{\mid }\le c$. The proofs are based on the hypergeometric method of Thue and Siegel and its refinement by Evertse.