  We study the dispersion of point sets in the unit square; i.e. the size of
the largest axes-parallel box amidst such point sets. It is known that
$\liminf_{N\to\infty} N\mathrm{disp}(N,2)\in \left[\frac54,2\right],$ where
$\mathrm{disp}(N,2)$ is the minimal possible dispersion for an $N$-element
point set in the unit square. The upper bound 2 is obtained by an explicit
point construction - the well-known Fibonacci lattice. In this paper we find a
modification of this point set such that its dispersion is significantly lower
than the dispersion of the Fibonacci lattice. Our main result will imply that
$\liminf_{N\to\infty} N\mathrm{disp}(N,2)\leq \varphi^3/\sqrt{5}=1.894427...$
