Additive Relations in Fields: An Entropy Approach

Let x1,...,xr be complex numbers with K = Q(x1,...,xr) having transcendence degree r-1 over Q. Consider the equation a1x1+...+akxk = 1, (1) in which the ai's are fixed elements of K×, no proper subsum ai1xi1+...+aijxij vanishes, and we seek solutions xi Î G = . It is well-known that (1) has only finitely many solutions; we present here an elementary proof of this fact using results from the entropy theory of commuting group automorphisms.