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Using techniques developed by Bourgain and Goldstein [\\textit{{Ann. of Math. 152(3):8"},"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":"1711.08661","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.SP","submitted_at":"2017-11-23T11:43:31Z","cross_cats_sorted":["math-ph","math.DS","math.MP"],"title_canon_sha256":"a74c33acbcf68ff76812f89ea7fd4143a4831f6b67c923dacf29c47c036a0395","abstract_canon_sha256":"2917ff46c3488f295623352eab431a244fd3e053e543a280f6c1393099083e5d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:06:25.365574Z","signature_b64":"TEx398oVgK5r34fU6nt2RxrTZACKAOJymMjjQmn2gn3508yKN9FExzKCdjH4We8WqseO5cO1R3KgAnyqpnCSAw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"49b9f5687c3de0bd067d719603a61c4bbc85037fad43ba85b0b9e7ab93191afe","last_reissued_at":"2026-05-18T00:06:25.364859Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:06:25.364859Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Anderson localization for one-frequency quasi-periodic block operators with long-range interactions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.DS","math.MP"],"primary_cat":"math.SP","authors_text":"Wenwen Jian, Xiaoping Yuan, Yunfeng Shi","submitted_at":"2017-11-23T11:43:31Z","abstract_excerpt":"In this paper, we study the quasi-periodic operators $H_{\\epsilon,\\omega}(x)$: $$(H_{\\epsilon,\\omega}(x)\\vec{\\psi})_n=\\epsilon\\sum_{k\\in\\mathbb{Z}}W_k\\vec{\\psi}_{n-k}+V(x+n\\omega)\\vec{\\psi}_n,$$ where $$\\vec{\\psi}=\\{\\vec{\\psi}_n\\}\\in\\ell^2(\\mathbb{Z},\\mathbb{C}^l),\\ V(x)=\\text{diag}\\left(v_1(x),\\cdots,v_l(x)\\right)$$ with $v_i$ ($1\\leq i \\leq l$) being real analytic functions on $\\mathbb{T}=\\mathbb{R}/\\mathbb{Z}$ and $W_k$ ($k\\in\\mathbb{Z}$) being $l\\times l$ matrices satisfying $\\|W_k\\|\\leq C_0e^{-\\rho|k|}$. 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