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Firstly we investigate the sums \\[ S_N(\\alpha,\\gamma):=\\sum_{n=1}^N\\frac{1}{n\\|n\\alpha-\\gamma\\|}~~~\\text{and}~~~ R_N(\\alpha,\\gamma):=\\sum_{n=1}^N\\frac{1}{\\|n\\alpha-\\gamma\\|}\\,, \\] where $\\alpha$ and $\\gamma$ are real parameters and $\\|\\cdot\\|$ is the distance to the nearest integer. Our theorems improve upon previous results of W. M. Schmidt and others, and are (up to constants) best possible. Related to the above sums, we also obtain upper and lower bounds for the cardinality of \\[ \\{1\\le n\\le N:\\|n\\alpha-\\gamma\\|<\\varepsilon\\} \\, , \\] vali"},"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":"1511.06862","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-11-21T09:50:08Z","cross_cats_sorted":[],"title_canon_sha256":"5a0380d5290d7dd5616580be140e81457a46f3be47b36b4f8902d2103fc48e52","abstract_canon_sha256":"f0ab03c13ecc40b7f29e06f538fa9b8da34be4f141e7c0e599743dd654e511b2"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:47:29.448222Z","signature_b64":"+M7N9wNXzac/8FO/qK5ISZ+7RQTeGXnAZ3qFiqp1k5uJaFXkDYtR++HjWtXb1bvXlov1sT9atGBkXRktaZCcAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"b8b35d1a2a6c524519adc9ea1759bc3383d1bcf1d08bda66cfe3614b433bf2f6","last_reissued_at":"2026-05-18T00:47:29.447784Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:47:29.447784Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Sums of reciprocals of fractional parts and multiplicative Diophantine approximation","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Alan Haynes, Sanju Velani, Victor Beresnevich","submitted_at":"2015-11-21T09:50:08Z","abstract_excerpt":"There are two main interrelated goals of this paper. 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