= Syndetic set
{wiki}
A <subset> $S\subseteq\mathbb Z$ is syndetic if it has bounded gaps: some positive <integer> $L$ makes every block $\{n,n+1,\ldots,n+L-1\}$ 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 $\{2^k,\ldots,2^k+k-1\}$ for $k\geq1$. The removed blocks have unbounded lengths, but at most $\sum_{k\leq\log_2N}k=O((\log N)^2)$ 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.
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