Sang June Lee (이상준), On strong Sidon sets of integers

Let $\mathbb N$ be the set of natural numbers. A set $A\subset \mathbb N$ is called a Sidon set if the sums $a_1+a_2$, with $a_1,a_2\in S$ and $a_1\leq a_2$, are distinct, or equivalently, if \[|(x+w)-(y+z)|\geq 1\] for every $x,y,z,w\in S$ with $x<y\leq z<w$. We define strong Sidon sets as follows:

For a constant $\alpha$ with $0\leq \alpha<1$, a set $S\subset \mathbb N$ is called an $\alpha$-strong Sidon set if \[|(x+w)-(y+z)|\geq w^\alpha\] for every $x,y,z,w\in S$ with $x<y\leq z<w$.

The motivation of strong Sidon sets is that a strong Sidon set generates many Sidon sets by altering each element a bit. This infers that a dense strong Sidon set will guarantee a dense Sidon set contained in a sparse random subset of $\mathbb N$.

In this talk, we are interested in how dense a strong Sidon set can be. This is joint work with Yoshiharu Kohayakawa, Carlos Gustavo Moreira and Vojtěch Rödl.