  The systolic ratio of a contact form $\alpha$ on the three-sphere is the
quantity \[ \rho_{\mathrm{sys}}(\alpha) =
\frac{T_{\min}(\alpha)^2}{\mathrm{vol}(S^3,\alpha\wedge d\alpha)}, \] where
$T_{\min}(\alpha)$ is the minimal period of closed Reeb orbits on
$(S^3,\alpha)$. A Zoll contact form is a contact form such that all the orbits
of the corresponding Reeb flow are closed and have the same period. Our first
main result is that $\rho_{\mathrm{sys}}\leq 1$ in a neighbourhood of the space
of Zoll contact forms on $S^3$, with equality holding precisely at Zoll contact
forms. This implies a particular case of a conjecture of Viterbo, a local
middle-dimensional non-squeezing theorem, and a sharp systolic inequality for
Finsler metrics on the two-sphere which are close to Zoll ones. Our second main
result is that $\rho_{\mathrm{sys}}$ is unbounded from above on the space of
tight contact forms on $S^3$.
