Sidon sets and sum-product phenomena
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Abstract: Given natural numbers s and k, we say that a finite set X of integers is an additive B_s[k] set if for any integer n, the number of solutions to the equation n=x_1+...+x_s, with x_1,..., x_s lying in X, is at most k, where we consider two such solutions to be the same if they differ only in the ordering of the summands. We define a multiplicative B_s[k] set analogously. These sets have been studied thoroughly from various different perspectives in combinatorial and additive number theory. In this talk, we will discuss sum-product phenomena for these sets. We show that there is either an exceptionally large additive B_s[k] set, or an exceptionally large multiplicative B_s[k] set, with k<<s. This is joint work with Akshat Mudgal.
