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We prove that for any $\\delta>0$ there is $L>0$ depending only on $\\delta$, and a subset $\\mathcal{N}$ of $B_2^n$ of cardinality at most $\\exp(\\delta n)$ such that with probability very close to one we have $$A(B_2^n)\\subset \\bigcup_{y\\in A(\\mathcal{N})}\\bigl(y+L\\sqrt{n}B_2^n\\bigr).$$ As an application, we show that for some $L'>0$ and $u\\in[0,1)$ depending only on the distribution law of $a_{11}$, the smallest singular value $s_n$ of the matrix $A$ satisfies "},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1508.06690","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.PR","submitted_at":"2015-08-26T23:56:06Z","cross_cats_sorted":[],"title_canon_sha256":"4c2ddc805aa62c91d68e5b133e945b9f94297ef54c50f1eb6ba6431b8cbe4d64","abstract_canon_sha256":"4170b1558edd568e05f226f39479bcd346b8ca7ebae30172dd086d63792bc3ac"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:50:43.050092Z","signature_b64":"kY/N3X+PfxEC049/G5cAylxHwiW3gPBFosakY7zKNIbCcnQ5mM3kDonMVfhfvMBHB22H2nrTAGwQW/G+ua4TBg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"f16ad7cad53785e1f25451ed6798bf8327c93bd4d51c945d2a93df64563b23a3","last_reissued_at":"2026-05-18T00:50:43.049600Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:50:43.049600Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Coverings of random ellipsoids, and invertibility of matrices with i.i.d. heavy-tailed entries","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.PR","authors_text":"Elizaveta Rebrova, Konstantin Tikhomirov","submitted_at":"2015-08-26T23:56:06Z","abstract_excerpt":"Let $A=(a_{ij})$ be an $n\\times n$ random matrix with i.i.d. entries such that $\\mathbb{E} a_{11} = 0$ and $\\mathbb{E} {a_{11}}^2 = 1$. 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