Research Projects

In transcendental number theory, one typically studies the arithmetic of elements of a non-empty subset A of complex numbers. In this regard, one tries to answer the following questions: (i) does this set A contain at least one transcendental number? (ii) are the elements of this set linearly independent over Q? (iii) are the elements of this set algebraically independent? There are very few techniques to deal with such problems. One of the powerful methods is diophantine approximation, which deals with approximation of irrational numbers by rational numbers. In fact, Liouville used rational approximations to prove the existence of transcendental numbers. This area of mathematics has a rich history and one of the major milestones is a famous result of Roth, known as Roth’s theorem (or Thue-Roth’s theorem). Schmidt proved a multidimensional extension of Roth's theorem, which is known as Subspace Theorem. Using this theorem, Corvaja and Zannier proved approximations of an algebraic number by the product of rational numbers and S-units. PI recently proved generalization of their theorem and independently an inhomogeneous analogue of their theorem. Our first goal is to extend this result beyond the setting of S-units and investigate its applications. The study of the sequence of fractional parts of powers of a real number greater than 1 is an interesting topic in diophantine approximation. It is not known whether the fractional parts of e^n tend to infinity or not. A folklore conjecture states that there is no transcendental number greater than 1 for which the sequence of its fractional parts of powers converges to 0. Rational approximations of powers of an algebraic number play a crucial role in studying the fractional parts of powers and continued fraction of powers of a real algebraic number. Corvaja and Zannier classified all algebraic numbers greater than 1 whose sequence of fractional parts of powers has a subsequence that converges exponentially to 0 or 1, this was a long standing problem of Mahler. They also showed that the period length of the continued fractions of powers of a quadratic irrational tends to infinity. Our second goal is to study the rational approximation to linear combinations of powers, and investigate interesting results on continued fraction. In 1929, Mahler developed a new method to prove the transcendence of the values of certain analytic functions at algebraic points that satisfy functional equations. Later, his method extended to prove the algebraic independence of values of such functions. Corvaja and Zannier proved the transcendence of more general lacunary type series as an application of the Subspace Theorem, recovering Mahler's result as a special case. Their approach did not rely on functional equations. Our final aim of this project is to explore their method to address questions of linear or algebraic independence for more general lacunary type series that do not satisfy functional equations.