Estimates of lengths of shortest nonzero vectors in some lattices. I

In 1988, Friese et al. put forward lower estimates for the lengths of shortest nonzero vectors for “almost all” lattices of some families in the dimension 3. In 2004, the author of the present paper obtained a similar result for the dimension 4. Here we give a better estimate for the cardinality of the set of exceptional lattices for which the above estimates are not valid. In the case of dimension 4 we improve the upper estimate for an arbitrary chosen parameter that controls the accuracy of these lower estimates and for the number of exceptions. In this (first) part of the paper, we also prove some auxiliary results, which will be used in the second (main) part of the paper, in which an analogue of A. Friese et al. result will be given for dimension 5.