Lacunarity and Diophantine Approximations

Diophantine equations are among the oldest objects of study in mathematics. Already Pythagoras studied numbers such that the sum of squares of two of them is equal to the square of the third number. This question can be formulated as a Diophantine equation x^2+y^2=z^2. It is oftendifficulttosolve Diophantineequations.Ofmy interest are various questions about solutions to Diophantine equations. I want to study them through contemporary techniques. I want to do so jointly with researchers at theUniversityof British Columbia (MichaelBennett, Dragos Ghioca, etc.) and with researchers at TU Graz and University of Salzburg (Clemens Fuchs, Robert Tichy, etc.) Many historicallyinteresting Diophantine equationsare of type f(x)=g(y), where both f and g have a fixed number of terms, and in particular when they have ``few`` terms. We call such f and g lacunary. Lacunary polynomials have been studied from various viewpoints. I am interested in their behavior with respect to functional composition. Such questions for arbitrary polynomials, as well as applications to various areas of mathematics, have been studied by many researchers, starting with an American mathematician J.F. Ritt in the 1920`s. Ritt`s results are considered to be fundamental. I want to apply new methods (arising from my work on compositions of covers of curves, which can be seen as objects generalizing rational functions) to gain further insights about lacunary polynomials. Of my interest are also connections to other areas of mathematics, such as complex analysis, arithmetic dynamics, etc. I am interested in contemporary methods for solving Diophantine equations. The ability to solve Diophantine equations has significantly improved in recent years, due to improvements in theory and development of computational tools. In the case of equations of type f(x)=g(y), where both f and g have ``few`` terms, I seek sharp results about the number of solutions, or I aim at completely solving them. Such questions are of centralinterestto number theorists. Numbertheorynowadays has applications to various areas of mathematics, and to modern cryptography.