Masters Thesis

The shape of domains of approximation

The Domain of Approximation is a nonconvex piecewise linear domain that is particularly useful for studying the Littlewood Conjecture, an open problem in Number Theory that involves the simultaneous approximation of real numbers. The Littlewood Conjecture, proposed by John Littlewood in the 1930s, states the following: for any real numbers a and /3, lim infn||na||||nj3|| = 0 as n approaches infinity where || • || denotes the distanceto the nearest integer. My work involves a geometric interpretation of the simultaneous approximation of real numbers in higher dimensions. From continued fraction theory, a convergent of a real number a is the fraction obtained by truncating a 's continued fraction. For example, ^ is the convergent that some use to represent n. I generalize the definition of a convergent for J-tuples in Rd and use geometric representations.

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