Restricted Successive Minima

TL;DR

This paper establishes bounds on the successive minima of symmetric convex bodies when lattice points are restricted from lying in certain forbidden sublattices, extending previous results to full-dimensional cases and all minima.

Contribution

It introduces new bounds for successive minima under lattice restrictions, broadening the scope to include full-dimensional forbidden lattices and all successive minima.

Findings

Extended bounds to full-dimensional forbidden lattices

Covered all successive minima cases

Complemented previous lower-dimensional results

Abstract

We give bounds on the successive minima of an $o$-symmetric convex body under the restriction that the lattice points realizing the successive minima are not contained in a collection of forbidden sublattices. Our investigations extend former results to forbidden full-dimensional lattices, to all successive minima and complement former results in the lower dimensional case.