| Summary: | This thesis includes three papers and one expository chapter as background
for one of the papers. These papers have in common that they combine
algebra with discrete geometry, mostly by using algebraic tools to prove
statements from discrete geometry. Algebraic curves and number theory
also recur throughout the proofs and results. In Chapter 1, we will detail
these common threads.
In Chapter 2, we prove that an infinite set of points in R² such that all
pairwise distances are rational cannot be contained in an algebraic curve,
except if that curve is a line or a circle, in which case at most 4 respectively 3
points of the set can be outside the line or circle. In the proof we use the
classification of curves by their genus, and Faltings' Theorem.
In Chapter 3, we informally present an elementary method for computing
the genus of a planar algebraic curve, illustrating some of the techniques in
Chapter 2.
In Chapter 4, we prove a bound on the number of unit distances that can
occur between points of a finite set in R², under the restriction that the line
segments corresponding to these distances make a rational angle with the
horizontal axis. In the proof we use graph theory and an algebraic theorem
of Mann.
In Chapter 5, we give an upper bound on the length of a simultaneous
arithmetic progression (a two-dimensional generalization of an arithmetic
progression) on an elliptic curve, as well as for more general curves. We
give a simple proof using a theorem of Jarnik, and another proof using the
Crossing Inequality and some bounds from elementary algebraic geometry,
which gives better explicit bounds.
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