Families of Unit Equations and Exponential Diophantine Problems via Integral Points

Abstract

This paper investigates the distribution of integral points on projective varieties via two distinct methods: the Ru-Vojta theorem and our higher-dimensional generalization of the Huang-Levin-Xiao inequalities. These approaches operate under distinct geometric conditions, specifically the transverse and proper intersections of boundary divisors. Applying this framework, we prove degeneracy results for the solution sets of two classes of one-parameter families of unit equations, differentiated by the degrees of their polynomial coefficients. Finally, we extend previous greatest common divisor (GCD) estimates to derive new results for specific exponential Diophantine equations and the distribution of digits in $q$-adic representations.