Adiceam, Faustin
(2014)
A Contribution to Metric Diophantine
Approximation :
the Lebesgue and Hausdorff Theories.
PhD thesis, National University of Ireland Maynooth.
Abstract
This thesis is concerned with the theory of Diophantine approximation from the point of
view of measure theory. After the prolegomena which conclude with a number of conjectures set
to understand better the distribution of rational points on algebraic planar curves, Chapter 1
provides an extension of the celebrated Theorem of Duffin and Schaeffer. This enables one to
set a generalized version of the Duffin–Schaeffer conjecture. Chapter 2 deals with the topic of
simultaneous approximation on manifolds, more precisely on polynomial curves. The aim is
to develop a theory of approximation in the so far unstudied case when such curves are not
defined by integer polynomials. A new concept of so–called “liminf sets” is then introduced in
Chapters 3 and 4 in the framework of simultaneous approximation of independent quantities.
In short, in this type of problem, one prescribes the set of integers which the denominators of
all the possible rational approximants of a given vector have to belong to. Finally, a reasonably
complete theory of the approximation of an irrational by rational fractions whose numerators
and denominators lie in prescribed arithmetic progressions is developed in chapter 5. This
provides the first example of a Khintchine type result in the context of so–called uniform
approximation.
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