Chapter 1: Small Doubling
نویسنده
چکیده
To get some intuition, we begin with the group of real numbers R under addition. A trivial upper bound for the size of a sum set is |A + A| ≤ |A| 2+|A| 2 (equality is obtained when every pair of elements of A have a distinct sum). If we build A by taking elements of R at random, then we expect |A+A| to be very close to this upper bound, since the probability of a1+a2 = a3+a4 is very small (where a1, a2, a3, a4 ∈ A). On the other hand, if A = {1, 2, . . . , n} then |A+ A| = |{2, 3, 4, . . . , 2n}| = 2|A| − 1. The same bound holds whenever A is an arithmetic progression. One of the main problems of additive combinatorics is characterizing the finite sets A (in either R or some other group) for which |A + A| is small with respect to |A|. As a warmup, we begin with the following easy claim.
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Inflation of Finite Lattices along All-or-nothing Sets a Dissertation Submitted to the Graduate Division of the University of Hawai‘i at Mānoa in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy
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