No description available.

Roth's theorem, proved by Roth in 1953, states that when A is a subset of the integers [1,N] with A dense enough, A has a three term arithmetic progression (3-AP). Since then the bound originally given by Roth has been improved upon by number theorists several times. The theorem can also be generalized to finite abelian groups. In 1994 Meshulam worked on finding an upper bound for subsets containing only trivial 3-APs based on the number of components in a finite abelian group. Meshulam’s bound holds for finite abelian groups of odd order. In 2003 Lev generalised Meshulam’s result for almost all finite abelian groups. In 2009 Liu and Spencer generalised the concept of a 3-AP to a linear equation and obtained a similar bound depending on the number of components of the group. In 2011, Liu, Spencer and Zhao generalised the 3-AP to a system of linear equations. This thesis is an overview of these results.

Using a classical result of Thue, we give an upper bound for the number of solutions to a family of quartic Thue equations. We also give an upper bound upon the number of solutions to a family of quartic Thue inequalities. Using the Thue-Siegel principle and the theory of linear forms in logarithms, an upper bound is given for general quartic Thue equations. As an application of the method of Thue-Siegel, we will resolve a conjecture of Walsh to the effect that the Diophantine equation aX⁴ - bY² = 1, for fixed positive integers a and b, possesses at most two solutions in positive integers X and Y. Since there are infinitely many pairs (a, b) for which two such solutions exist, this result is sharp. It is also effectively proved that for fixed positive integers a and b, there are at most two positive integer solutions to the quartic Diophantine equation aX⁴ - bY² = 2. We will also study cubic and quartic Thue equations by combining some classical methods from Diophantine analysis with modern geometric ideas.

Thesis (Ph. D.)--McMaster University (Canada), 1996. / Source: Dissertation Abstracts International, Volume: 58-06, Section: B, page: 3083. Adviser: V.P. Snaith.

Diss. Uppsala : Univ., 2005.

Thesis (Ph.D.)--University of Adelaide, School of Mechanical Engineering, 2006. / Includes author's previously published papers. "June 2006" Bibliography: p. 215-220. Also available in print form.

Thesis (Ph. D.)--Virginia Polytechnic Institute and State University, 1992. / Vita. Abstract. Includes bibliographical references (leaves 75-79). Also available via the Internet.

Thesis (M.Sc.) - Carleton University, 2002. / Includes bibliographical references (p. 226-228). Also available in electronic format on the Internet.

Thesis (M. Sc.)--Carleton University, 2003. / Includes bibliographical references (p. 143-145). Also available in electronic format on the Internet.

Thesis (Ph. D.)--Columbia University. / "Being vol. XIII, no. 1, of Columbia university contributions of philosophy, psychology and education. The results, of this research were presented before the Section of anthropology and psychology of the New York academy of sciences, and the monograph is published under the auspices of the Academy."