{"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. Conversely, it is known that $E\\left(A_{N}\\right)=\\mathcal{O}\\left(N^{3-\\varepsilo"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1708.08599","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}