Abstract: | In the first part of this thesis, we present the first non-trivial small value estimate that applies to an algebraic group of dimension 2 and which involves large sets of points. The algebraic group that we consider is the product ℂ× ℂ*, of the additive group ℂ by the multiplicative group ℂ*. Our main result assumes the existence of a sequence (PD)D ≥1 of non-zero polynomials in ℤ [X1, X2] taking small absolute values at many translates of a fixed point (ξ, η) in ℂ × ℂ* by consecutive multiples of a rational point (r, s) ∈ (ℚ*)2 with s = ±1. Under precise conditions on the size of the coefficients of the polynomials PD, the number of translates of (ξ, η) and the absolute values of the polynomials PD at these points, we conclude that both ξ and η are algebraic over ℚ. We also show that the conditions that we impose are close from being best possible upon comparing them with what can be achieved through an application of Dirichlet’s box principle.
In the second part of the thesis, we consider points of the form θ = (1,θ1 , . . . ,θd-1 ,ξ) where {1,θ1 , . . . ,θd-1 } is a basis of a real number field K of degree d ≥ 2 over ℚ and where ξ is a real number not in K. Our main results provide sharp upper bounds for the uniform exponent of approximation to θ by rational points, denoted λ ̂(θ), and for its dual uniform exponent of approximation, denoted τ ̂(θ). For d = 2, these estimates are best possible thanks to recent work of Roy. We do not know if they are best possible for other values of d. However, in Chapter 2, we provide additional information about rational approximations to such a point θ assuming that its exponent λ ̂(θ) achieves our upper bound. In the course of the proofs, we introduce new constructions which are interesting by themselves and should be useful for future research. |