### Finding large additive and multiplicative Sidon sets in sets of integers.

Combinatorics Seminar

15th March 2022, 11:00 am – 12:00 pm

Fry Building, LG.02 (note the non-standard room)

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. For instance, even in the case s=2 and k=1 (where such sets are referred to as Sidon sets), the problem of characterising the largest additive B_{s}[k] set in {1, 2, ..., N} remains a major open question in the area.

In this talk, we consider this problem from an arithmetic combinatorial perspective, and so, we show that for every natural number s and for every finite set A of integers, A either contains an additive B_{s}[1] set B or a multiplicative B_{s}[1] set C satisfying

max{ |B| , |C| } >> |A|^{c/s},

where c >> (log log s)^{1/2 - o(1)}.

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