Auxiliary Functions From The Pigeonhole Principle
The auxiliary functions sketched above can all be explicitly calculated and worked with. A breakthrough by Axel Thue and Carl Ludwig Siegel in the twentieth century was the realisation that these functions don't necessarily need to be explicitly known – it can be enough to know they exist and have certain properties. Using the Pigeonhole Principle Thue, and later Siegel, managed to prove the existence of auxiliary functions which, for example, took the value zero at many different points, or took high order zeros at a smaller collection of points. Moreover they proved it was possible to construct such functions without making the functions too large. Their auxiliary functions were not explicit functions, then, but by knowing that a certain function with certain properties existed, they used its properties to simplify the transcendence proofs of the nineteenth century and give several new results.
This method was picked up on and used by several other mathematicians, including Alexander Gelfond and Theodor Schneider who used it independently to prove the Gelfond–Schneider theorem. Alan Baker also used the method in the 1960s for his work on linear forms in logarithms and ultimately Baker's theorem. Another example of the use of this method from the 1960s is outlined below.
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