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New quantitative estimates on the incidence geometry and growth of finite sets

This thesis establishes new quantitative records in several problems of incidence geometry and growth. After the necessary background in Chapters 1, 2 and 3, the following results are proven. Chapter 4 gives new results in the incidence geometry of a plane determined by a finite field of prime order. These comprise a new upper bound on the total number of incidences determined by finitely many points and lines, and a new estimate for the number of distinct lines determined by a finite set of non-collinear points. Chapter 5 gives new results on expander functions. First, a new bound is established for the two-variable expander a+ab over a finite field of prime order. Second, new expanders in three and four variables are demonstrated over the real and complex numbers with stronger growth properties than any functions previously considered. Finally, Chapter 6 gives the first bespoke sum-product estimate over function fields, a setting that has so far been largely unexplored for these kinds of problems. This last chapter is joint work with Thomas Bloom.