{"paper":{"title":"On the number of gaps of sequences with Poissonian Pair Correlations","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CA"],"primary_cat":"math.NT","authors_text":"Christoph Aistleitner, Paolo Leonetti, Paolo Minelli, Thomas Lachmann","submitted_at":"2019-08-17T10:51:45Z","abstract_excerpt":"A sequence $(x_n)$ on the torus is said to have Poissonian pair correlations if $\\# \\{1\\le i\\neq j\\le N: |x_i-x_j| \\le s/N\\}=2sN(1+o(1))$ for all reals $s>0$, as $N\\to \\infty$.\n  It is known that, if $(x_n)$ has Poissonian pair correlations, then the number $g(n)$ of different gap lengths between neighboring elements of $\\{x_1,\\ldots,x_n\\}$ cannot be bounded along every index subsequence $(n_t)$. First, we improve this by showing that the maximum among the multiplicities of the neighboring gap lengths of $\\{x_1,\\ldots,x_n\\}$ is $o(n)$, as $n\\to \\infty$. Furthermore, we show that, for every fun"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1908.06292","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/1908.06292/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}