{"paper":{"title":"Some negative results related to Poissonian pair correlation problems","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Gerhard Larcher, Wolfgang Stockinger","submitted_at":"2018-03-14T12:30:19Z","abstract_excerpt":"We say that a sequence $(x_n)_{n \\in \\mathbb{N}}$ in $[0,1)$ has Poissonian pair correlations if \\begin{equation*} \\lim_{N \\to \\infty} \\frac{1}{N} \\# \\left \\lbrace 1 \\leq l \\neq m \\leq N: \\| x_l - x_m \\| \\leq \\frac{s}{N} \\right \\rbrace = 2s \\end{equation*} for every $s \\geq 0$. The aim of this article is twofold. First, we will establish a gap theorem which allows to deduce that a sequence $(x_n)_{n \\in \\mathbb{N}}$ of real numbers in $[0,1)$ having a certain weak gap structure, cannot have Poissonian pair correlations. This result covers a broad class of sequences, e.g., Kronecker sequences, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1803.05236","kind":"arxiv","version":2},"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"}