Upper Banach density

For $E\subseteq\mathbb Z$ , its upper Banach density is

d^*(E)=\limsup_{N\to\infty}\ \sup_{m\in\mathbb Z} \frac{|E\cap\{m,m+1,\ldots,m+N-1\}|}{N}.

It measures the greatest density that persists in arbitrarily long intervals, allowing their starting positions to vary.

Comparison with ordinary upper density

For $E\subseteq\mathbb N$ , the ordinary upper density is $\limsup_N |E\cap\{1,\ldots,N\}|/N$ , which cannot exceed $d^*(E)$ . The two can differ: a set containing arbitrarily long intervals separated by sufficiently large gaps has upper Banach density one but can have ordinary upper density zero.

Both positive-density notions imply the existence of arbitrarily long arithmetic progressions by Szemerédi's theorem.