Discrete Analysis Seminar

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 $k=1$, wherein 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, the largest additive $B_{s}[1]$ subset $B$ of $A$ and the largest multiplicative $B_{s}[1]$ subset $C$ of $A$ satisfy

\[ \max \{ |B| , |C| \} \gg_s |A|^{\eta_s/s} , \]

where $\eta_s \gg (\log \log s)^{1/2 - o(1)}$.