{"paper":{"title":"Convergence results for systems of linear forms on cyclic groups, and periodic nilsequences","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Olof Sisask, Pablo Candela","submitted_at":"2012-12-15T13:13:27Z","abstract_excerpt":"Given a positive integer $N$ and real number $\\alpha\\in [0, 1]$, let $m(\\alpha,N)$ denote the minimum, over all sets $A\\subset \\mathbb{Z}/N\\mathbb{Z}$ of size at least $\\alpha N$, of the normalized count of 3-term arithmetic progressions contained in $A$. A theorem of Croot states that $m(\\alpha,N)$ converges as $N\\to\\infty$ through the primes, answering a question of Green. Using recent advances in higher-order Fourier analysis, we prove an extension of this theorem, showing that the result holds for $k$-term progressions for general $k$ and further for all systems of integer linear forms of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1212.3681","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"}