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We prove that if the additive energy $E\\left(A_{N}\\right)$ of $A_{N}$ is in $\\Omega\\left(N^{3}\\right)$, then the sequence $\\left(\\left\\langle \\alpha a_{n}\\right\\rangle \\right)_{n}$ of fractional parts of $\\alpha a_{n}$ does not have Poissonian pair correlations (PPC) for almost every $\\alpha$ in the sense of Lebesgue measure. Conversely, it is known that $E\\left(A_{N}\\right)=\\mathcal{O}\\left(N^{3-\\varepsilo"},"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":"1708.08599","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2017-08-29T06:21:45Z","cross_cats_sorted":[],"title_canon_sha256":"de5e9694a71792400e1ad7872d88987313850e63dcec179fdd5dcd2b630d8f00","abstract_canon_sha256":"05ef49925bd2b6d1d5ac6e316cfb3c9cb07bb9f24196f2d54ed77f636fc7a962"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:36:27.304288Z","signature_b64":"B8kJ+gwGsMsfZXzxRcbik3JCbZ6EUftcQkbpoi5Oxt8llSXVejGtogWuIbXcr7VeolCaf9ypXmvbEA9Jmf4CAg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3c0f2fa2396ed248d0211d3f06368473be947a8ca2ce6581bcb8df615d61f573","last_reissued_at":"2026-05-18T00:36:27.303572Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:36:27.303572Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Exceptional Sets in the Metric Poissonian Pair Correlations problem","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Niclas Technau, Thomas Lachmann","submitted_at":"2017-08-29T06:21:45Z","abstract_excerpt":"Let $\\left(a_{n}\\right)_{n}$ be a strictly increasing sequence of positive integers, denote by $A_{N}=\\left\\{ a_{n}:\\,n\\leq N\\right\\} $ its truncations, and let $\\alpha\\in\\left[0,1\\right]$. We prove that if the additive energy $E\\left(A_{N}\\right)$ of $A_{N}$ is in $\\Omega\\left(N^{3}\\right)$, then the sequence $\\left(\\left\\langle \\alpha a_{n}\\right\\rangle \\right)_{n}$ of fractional parts of $\\alpha a_{n}$ does not have Poissonian pair correlations (PPC) for almost every $\\alpha$ in the sense of Lebesgue measure. 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