A reverse Minkowski theorem

Oded Regev, Noah Stephens-Davidowitz

Research output: Contribution to journalArticlepeer-review


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 languageEnglish (US)
Pages (from-to)1-49
Number of pages49
JournalAnnals of Mathematics
Issue number1
StatePublished - 2024


  • Minkowski's theorem
  • geometry of numbers
  • lattices

ASJC Scopus subject areas

  • Mathematics (miscellaneous)


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