| In this dissertation, we investigate the relationships between and applications of several different notions of largeness for subsets of Z d, the d-dimensional integer lattice, and, more generally, groups and semigroups. The various meanings of the word "large" appearing here are inspired by ideas in algebra, analysis, and geometry and are significant for their historical influence and usefulness in applications to combinatorics and number theory.;This work is comprised of three main parts. In the first, we study the relationships between several naturally occurring notions of additive and multiplicative largeness in rings and semirings. We harness these relationships and the ideas behind them to give a number of novel applications at the intersection of Ramsey theory and number theory.;In the second part, we introduce the mass and counting measures and dimensions for subsets of Zd and explore the properties of large fractal subsets of the lattice. In particular, we prove analogues of Marstrand's projection theorem and derive applications concerning the size of sumsets of dilated subsets of Z. We then show how to derive the classic projection theorems of Marstrand and Kaufman, along with an open conjecture of Oberlin, from their discrete analogues.;In the third part, we study the existence of certain combinatorial configurations in members of a family of fractal subsets of the natural numbers, the Piatetski-Shapiro sets. We provide a metrical solution to the problem of determining which Piatetski-Shapiro sets contain infinitely many solutions to bivariate linear equations. |
