Published online by Cambridge University Press: 18 June 2010
In Chapter 2 we studied the elementary theory of sum sets A + B for general subsets A, B of an arbitrary additive group Z. In order to progress further with this theory, it is important first to understand an important subclass of such sets, namely those with a strong geometric and additive structure. Examples include (generalized) arithmetic progressions, convex sets, lattices, and finite subgroups. We will term the study of such sets (for want of a better name) additive geometry; this includes in particular the classical convex geometry of Minkowski (also known as geometry of numbers). Our aim here is to classify these sets and to understand the relationship between their geometrical structure, their dimension (or rank), their size (or volume, or measure), and their behavior under addition or subtraction. Despite looking rather different at first glance, it will transpire that progressions, lattices, groups, and convex bodies are all related to each other, both in a rigorous sense and also on the level of heuristic analogy. For instance, progressions and lattices play a similar role in arithmetic combinatorics that balls and subspaces play in the theory of normed vector spaces. In later sections, by combining methods of additive geometry, sum set estimates, Fourier analysis, and Freiman homomorphisms, we will be able to prove Freiman's theorem, which shows that all sets with small doubling constant can be efficiently approximated by progressions and similarly structured sets.
To save this book to your Kindle, first ensure [email protected] is added to your Approved Personal Document E-mail List under your Personal Document Settings on the Manage Your Content and Devices page of your Amazon account. Then enter the ‘name’ part of your Kindle email address below. Find out more about saving to your Kindle.
Note you can select to save to either the @free.kindle.com or @kindle.com variations. ‘@free.kindle.com’ emails are free but can only be saved to your device when it is connected to wi-fi. ‘@kindle.com’ emails can be delivered even when you are not connected to wi-fi, but note that service fees apply.
Find out more about the Kindle Personal Document Service.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Dropbox.
To save content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about saving content to Google Drive.