UBC Theses and Dissertations
Distance problems and points on curves : an algebraic approach Schwartz, Ryan
In this thesis we give results on unit and rational distances, structure results for surfaces containing many points of a cartesian product and a survey of various, mainly combinatorial, applications of the Subspace Theorem. We take an algebraic approach to most of the problems and use techniques from commutative algebra, incidence geometry, number theory and graph theory. In Chapter 2 we give extensions of a result of Elekes and Rónyai that says that if the graph of a real (or complex) polynomial contains many points of a cartesian product then the polynomial has a special additive or multiplicative form. We extend this to asymmetric cartesian products and to higher dimensions. Elekes and Rónyai’s result was used to prove a conjecture of Purdy on the number of distinct distances between two collinear point sets in the plane. Our extensions give improved bounds for the conjecture. In Chapter 3 we prove a special case of the Erdős unit distance problem. This problem asks for the maximum possible number of unit distances between n points in the plane in the form of an asymptotic upper bound. We provide an upper bound of n^(1+7/sqrt(log(n))) when we only consider unit distances corresponding to roots of unity and give a superlinear lower bound. We also consider related rational distance problems. We require an algebraic result of Mann on the number of solutions of linear equations of roots of unity. In Chapter 4 we extend our result from the previous chapter to unit distances coming from a multiplicative subgroup of ℂ^* of “low” rank. We use a corollary of the Subspace Theorem. In this case we get, for ε > 0, at most cn^(1+ε) unit distances. We show that the well known lower bound construction of Erdős for the general unit distance problem consists of distances from such a subgroup and so our result applies to the best known maximal unit distance sets. In Chapter 5 we give a survey of various applications of the Subspace Theorem including less well known combinatorial applications such as sum-product estimates and line configurations with few distinct intersections.
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