  The local theory of regular or multi-regular systems aims at finding
sufficient local conditions for a Delone set $X$ to be a regular or
multi-regular system. One of the main goals is to estimate the regularity
radius $\hat{\rho}_d$ for Delone sets $X$ in terms of the radius $R$ of the
largest "empty ball" for $X$.
  The present paper establishes the lower bound $\hat{\rho_d}\geq 2dR$ for all
$d$, which is linear in $d$. The best previously known lower bound had been
$\hat{\rho}_d\geq 4R$ for $d\geq 2$. The proof of the new lower bound is
accomplished through explicit constructions of Delone sets with mutually
equivalent $(2dR-\varepsilon)$-clusters, which are not regular systems.
