Abstract
We prove a conjecture due to Dadush, showing that if $$ is a lattice such that det$$ for all sublattices $$, then $$ where t := 10(log n+2). From this we derive bounds on the number of short lattice vectors, which can be viewed as a partial converse to Minkowski's celebrated first theorem. We also derive a bound on the covering radius.
| Original language | English (US) |
|---|---|
| Pages (from-to) | 1-49 |
| Number of pages | 49 |
| Journal | Annals of Mathematics |
| Volume | 199 |
| Issue number | 1 |
| DOIs | |
| State | Published - 2024 |
Keywords
- Minkowski's theorem
- geometry of numbers
- lattices
ASJC Scopus subject areas
- Mathematics (miscellaneous)