 Upper asymptotic density

For a subset $A$ of the positive integers, its upper asymptotic density is $\overline{d}(A)={\mathrm{limsup}}_{N\to \infty }|A\cap \{1,\dots ,N\}|/N$. Unlike natural density, it always exists. If natural density exists, the two agree. Positive upper asymptotic density suffices for the Szemerédi theorem.