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We consider the set $\\mathcal{S}=\\{\\langle {v}\\gamma,{w}\\rangle_{\\mathbb{R}^2}:\\gamma\\in\\Lambda\\}$, where $\\langle\\cdot,\\cdot\\rangle_{\\mathbb{R}^2}$ is the standard inner product in $\\mathbb{R}^2$. Using Hardy-Littlewood circle method and some infinite co-volume lattice point counting techniques developed by Bourgain, Kontorovich and Sarnak, together with Gamburd's 5/6 spectral gap, we show that if $\\Lambda$ has parabolic element"},"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":"1610.00770","kind":"arxiv","version":3},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-10-03T22:05:48Z","cross_cats_sorted":[],"title_canon_sha256":"398bad9ba719fc6de44e7c090145cde1440dc25467073d025e4f886448c40a5e","abstract_canon_sha256":"6c87eb8f5f62c3ef194a9cb4ecf7bb9555aa19f0c2c8ce4281291d4b163251c5"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:12:00.851369Z","signature_b64":"uKES66OxudhWFz6tY3tmlFHBC/J4ESAFyLAsV9UWYhH9iFKtHTxmb9hmoMA69VQqCxyey8X5CLfT8Luxj6oNDQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bf6c7558a15e33a4d11e48813d04c378ec31229c485272a2aff1690a7af90d41","last_reissued_at":"2026-05-18T00:12:00.850680Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:12:00.850680Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Representation of Integers from Thin Subgroups of SL(2,Z) with Parabolics","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Xin Zhang","submitted_at":"2016-10-03T22:05:48Z","abstract_excerpt":"Let $\\Lambda<SL(2,\\mathbb{Z})$ be a finitely generated, non-elementary Fuchsian group of the second kind, and $v, w$ be two primitive vectors in $\\mathbb{Z}^2-(0,0)$. 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