{"paper":{"title":"Partial difference equations over compact Abelian groups, I: modules of solutions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AC","math.CO","math.DS"],"primary_cat":"math.FA","authors_text":"Tim Austin","submitted_at":"2013-05-30T23:21:14Z","abstract_excerpt":"Consider a compact Abelian group $Z$ and closed subgroups $U_1$, \\ldots, $U_k \\leq Z$. Let $\\mathbb{T} := \\mathbb{R}/\\mathbb{Z}$. This paper examines two kinds of functional equation for measurable functions $Z\\to \\mathbb{T}$.\n  First, given $f:Z\\to \\mathbb{T}$ and $w \\in Z$, the resulting differenced function is \\[d_wf(z) := f(z-w) - f(z).\\] In this notation, we study solutions to the system of difference equations \\[d_{u_1}\\cdots d_{u_k}f \\equiv 0 \\quad \\forall u_1 \\in U_1,\\ u_2 \\in U_2,\\ \\ldots\\ u_k \\in U_k.\\] Second, we study tuples of measurable functions $f_i:Z\\to \\mathbb{T}$ such that $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1305.7269","kind":"arxiv","version":5},"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"}