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 $a_1\le a_2$, are distinct, or equivalently, if $$|(x+w)-(y+z)|\ge 1$$ for every $x,y,z,w\in S$ with $x<y\le z<w$. We define strong Sidon sets as follows:

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

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.