Research Interests

My research interests are diophantine approximation, diophantine equations, polynomials with restricted coefficients and merit factors of binary sequences.

Solving diophantine equations is one of the fundamental problems in number theory. Diophantine equations are equations with integer coefficients and integer unknowns. We are often interested in the number of solutions (is it finite or infinite), and the size of solutions of a diophantine equation. For example, Fermat's last problem (now a theorem), which concerns the existence of nontrivial solutions of the diophantine equation xⁿ+yⁿzⁿ, is, of course, one of these problems. Study of the solutions of diophantine equations in certain subsets of integers is also very interesting. For example, we can study the prime solutions of diophantine equations. One of my research work is solving certain types of diophantine equations with prime solutions, e.g. a₁p₁+a₂p₂+a₃p₃=b with prime solution p_j, by using analytic methods, such as Hardy and Littlewood method.

My second research interest is in studying of the growth for the norms of polynomials with integers coefficients. We are particularly interested in investigating the properties of polynomials having coefficients +1 or -1. We call these Littlewood polynomials. This class of polynomials has been studied from a number of points of view and a number of interesting open problems on these polynomials have been raised. We are interested in studying the behavior of the L₄ norms (over the unit circle) of Littlewood polynomials. More specifically, one may want to find a sequence of Littlewood polynomials having small L₄ norm. The present best example, known as the shifted Fekete polynomial, is constructed by using the Legendre symbol which is used to study the quadratic residue in number theory. One of our works proves some amazing exact formulas for L₄ norm of the shifted Fekete polynomials while all the previous results are just asymptotic estimations. We are now working on extending our ideas to estimate L₄ norm of other polynomials. This study is also related to the problem of finding the easily distinguishable binary sequences in the information and communication theory.

The followings are the other problems which I am recently interested in:

• Diophantine approximation in projective spaces over number fields.
• Cohn's problem on classifying characters over finite fields.
• Existence of planar functions.