Additive Combinatorics July 22, 2009Posted by lhstephens in Uncategorized.
My project at RSI is in additive combinatorics, which is a field of math, which extends the notions of set over operations. Here is a brief introduction to some of the terminology and theorems in the field.
Additive combinatorics studies subsets of abelian groups. To study these sets, additive combinatorics uses theorems and notation from both graph theory and set theory. An additive set with respect to an abelian group is a set . The group Z is referred to as an ambient group. Normally, the group is the integers over addition or .
The size of an additive set is the number of unique elements of and is denoted by . For example, if then . For two additive sets in an abelian group , the sumset of and is defined to be . For example, let and let then . Similarly the difference set, , is defined as .
The dilate of a set by , where is . This is different than which takes the form and is equivalent to with copies of . For example, if , then while . In particular, Bukh investigated sumsets of the form . Bukh proved that for any finite set , , where is a sharp error term. Then, Bukh proved that if either or then for .
The doubling constant of an additive set is defined as and is a measure of the number of pairs in with equal sums. The difference constant of an additive set is similarly defined as , and is a measure of the number of pairs with equal difference. Both the doubling constant and the difference constant are measures of the structure, or non-randomness, of an additive set .
Additive combinatorics also uses Graph theory and Fourier analysis to prove some of the more powerful theorems, for a more detailed look at the subject, see Terrence Tao’s and Van H. Vu’s book, Additive Combinatorics.