{"paper":{"title":"On the Minimum of a Positive Definite Quadratic Form over Non--Zero Lattice points. Theory and Applications","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.IT","math.IT"],"primary_cat":"math.NT","authors_text":"Evgeniy Zorin, Faustin Adiceam","submitted_at":"2016-07-15T11:29:56Z","abstract_excerpt":"Let $\\Sigma_d^{++}$ be the set of positive definite matrices with determinant 1 in dimension $d\\ge 2$. Identifying any two $SL_d(\\mathbb{Z})$-congruent elements in $\\Sigma_d^{++}$ gives rise to the space of reduced quadratic forms of determinant one, which in turn can be identified with the locally symmetric space $X_d:=SL_d(\\mathbb{Z})\\backslash SL_d(\\mathbb{R})/SO_d(\\mathbb{R})$. Equip the latter space with its natural probability measure coming from a Haar measure on $SL_d(\\mathbb{R})$. In 1998, Kleinbock and Margulis established sharp estimates for the probability that an element of $X_d$ "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1607.04467","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}