Masters Thesis

Understanding the complexity of the domain of approximation of a rational tuple

In continued fraction theory, a convergent is an irreducible rational number that best approximates a particular real number when compared to all rationals of smaller denominator. Generalizing this definition to include rational tuples of arbitrary length d shows that every point in Qd determines a domain of approximation: the set of all points in Rd that axe best approximated by this particular rational. It is known that the domain of approximation is rectilinear, with its sides parallel to the coordinate axes. We shall examine the geometric properties of a carefully constructed set of sheared lattices in order to characterize the "corner points" of the domain of approximation, with a particular emphasis placed on identifying "primary" corners. We then show that for any 2-tuple ( ^ , ^ ) € Q2, the number of primary corners in its corresponding domain of approximation is O(logg).

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