{"paper":{"title":"Carries and the arithmetic progression structure of sets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Francesco Monopoli, Imre Z. Ruzsa","submitted_at":"2015-06-29T21:22:41Z","abstract_excerpt":"If we want to represent integers in base $m$, we need a set $A$ of digits, which needs to be a complete set of residues modulo $m$. When adding two integers with last digits $a_1, a_2 \\in A$, we find the unique $a \\in A$ such that $a_1 + a_2 \\equiv a$ mod $m$, and call $(a_1 + a_2 -a)/m$ the carry. Carries occur also when addition is done modulo $m^2$, with $A$ chosen as a set of coset representatives for the cyclic group $\\mathbb{Z}/m \\mathbb{Z} \\subseteq \\mathbb{Z}/m^2\\mathbb{Z}$. It is a natural to look for sets $A$ which minimize the number of different carries. In a recent paper, Diaconis"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1506.08869","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"}