Research Projects

The problem of approximating algebraic numbers by rational numbers has a very rich history, starting with the works of Dirichlet and Liouville. One of the major milestones in this area is a famous result of Roth, known as Roth’s theorem. The multidimensional extension of Roth's theorem is due to Schmidt, and is known as the Subspace theorem. This is one of the deep results in Diophantine geometry, which has applications in various branches of number theory. In the last two decades several unexpected applications of this theorem were found. This includes a complete solution of a problem of Mahler on the behavior of nearest integer function of powers of algebraic numbers; Corvaja and Zannier achieved this by proving a Thue-Roth type inequality with moving target, where target set being a set of $S$-units. In this project we intend to primarily use the Subspace theorem as a central tool to study various problems in number theory. As a first project, we would like to prove a simultaneous approximation of several algebraic numbers by product of rational number and almost $(S,\delta)$-units. Recall that almost $(S,\delta)$-unit generalizes the definition of $S$-unit to more general algebraic number whose dominant factor in terms of height comes from the $S$-parts. In the direction of transcendence criteria, as an application of Subspace theorem, Corvaja and Zannier investigated the arithmetic nature of values of lacunary series at algebraic points, which includes the classical theorem of Mahler on Fredholm series. We intend to establish analogue results for the power series evaluated at almost $(S,\delta)$-units. As a second project, we would like to enumerate perfect powers among recurrence sequence. In 2006, Beguaed \textit{et al.} confirm the long-standing conjecture that the only Fibonacci numbers which are perfect powers are $F_0=0, F_1=F_2=1, F_6=8~ \mbox{and}~ F_{12}=144.$ Later, in 2021, Kebli \textit{et al.} solved $F_n\pm F_m=y^a$ under the $abc$ conjecture. Since Fibonacci sequence is a special kind of recurrence sequence, we intend to extend these results to general linear recurrence sequence. As a third project, we would like to study the behavior of continued fraction expansion of certain type of sequence. By an application of approximation of power of algebraic numbers by rational numbers, when $\alpha$ is a real quadratic irrational which is neither the square root of a rational number nor a unit, Corvaja and Zannier proved that period length of continued fraction expansion of $\alpha^n$ tends to infinity as $n\rightarrow\infty.$ Recently, Kumar, Singh and Sprang studied the period lengths of linear recurrence sequences which includes Corvaja and Zannier result. For a unbounded sequence $(a_n)$ and for special type of algebraic integers $\alpha_i,\beta_i,$ we intend to study period length of continued fraction expansion of $\lambda_1\frac{\alpha_1^{na_n}}{\alpha_1^n+\beta}+\cdots+\lambda_k\frac{\alpha_k^{na_n}}{\alpha_k^n+\beta}.$