| In this thesis I present solutions to four extremal problems in additive combinatorics and combinatorial geometry.;First, I present a result on the maximal density of sets of integers in which no two elements differ by an element of a prescribed set D, and show how its analogue for measurable sets in Rd (d ≥ 2) implies a result of Furstenberg-Katznelson-Weiss on distances in sets of positive density.;Second, I establish sharp bounds on the sizes of sums of dilates of the form lambda1 · A + ˙ ˙ ˙ + lambdak · A for a finite set A ⊂ Z .;Third, I use the upper bounds on the sums of dilates to derive the optimal bounds on the size of a subset of {1, ..., n} not containing solutions to a symmetric linear equation, resolving a question of Ruzsa.;Finally, I present the first non-trivial lower bounds on the size of weak epsilon-nets for convex sets. |
