BOUNDS FOR LEAST RELATIVE VACANCY IN A SIMPLE MOSAIC PROCESS

Let 0 < d < m < 1 and consider the mosaic process formed by centering d x d squares on the points of a Poisson process of intensity lambda i n the unit square D subset of or equal to R(2). If G denotes the union of these squares, then least relative vacancy is the infimum of the q uantity m(-2) x \S boolean AND G(c)\ taken over all m x m squares S su ch that S subset of or equal to D. The authors prove two-sided bounds for the distribution of least relative vacancy and show that the bound s are asymptotically sharp, in the logarithmic sense, as d tends to ze ro.