Thickly syndetic set

Question

Somewhere I read that, S is thickly syndetic if for every natural number N the positions where length N runs from a syndetic set. I can't understand this definition. Please help me to understand this definition or give me the references where I can find about this. Thank you.

Answer 1

The phrasing in your definition is slightly wrong. Let me fix it: a set $S$ of natural numbers is thickly syndetic, if for every natural number $n$, the positions of length-$n$ runs in $S$ form a syndetic set.

A length-$n$ run in $S$ is just an interval $I = \{i, i+1, i+2, \ldots, i+n-1\}$ of $n$ consecutive numbers that are all in $S$. If $S$ is thickly syndetic, then the set of the numbers $i$ that start length-$n$ runs in $S$ must be syndetic, separately for every $n$. This means that for some $k$, in every length-$k$ interval of natural numbers you can find a position of a length-$n$ run in $S$.

More formally: a set $S$ is thickly syndetic if for every $n \in \mathbb{N}$, there exists $k \in \mathbb{N}$ such that for every $p \in \mathbb{N}$, there exists $i \in \{p, p+1, \ldots, p+k-1\}$ such that $\{i, i+1, \ldots, i+n-1\} \subset S$.