 Syndetic set

A subset $S\subseteq \mathbb{Z}$ is syndetic if it has bounded gaps: some positive integer $L$ makes every block $\{n,n+1,\dots ,n+L-1\}$ meet $S$. The quantifier applies at every integer location, including negative ones. A corresponding one-sided definition uses $S\subseteq \mathbb{N}$ and nonnegative starting locations.

For example, the even integers are syndetic with $L=2$, while the positive perfect squares are not: the gap between $m^2$ and $(m+1{)}^2$ is $2m+1$ and grows without bound. In a one-sided syndetic set, partitioning the first $N$ integers into blocks of length $L$ gives at least $\lfloor N/L\rfloor$ elements, so its lower asymptotic density is at least $1/L$.

Positive lower density alone does not ensure bounded gaps. Remove from $\mathbb{N}$ each block $\{2^k,\dots ,2^k+k-1\}$ for $k\ge 1$. The removed blocks have unbounded lengths, but at most ${\sum }_{k\le {\log }_{2}N}k=O((\log N{)}^2)$ removed points lie below $N$. The remaining set has asymptotic density one and is not syndetic. Thus a condition on every interval is stronger than a condition on the long-run proportion.

In mathematics, a syndetic set is a type of subset of the integers or natural numbers that is characterized by the property of having bounded gaps between its elements.