{"paper":{"title":"On the cardinality of general $h$-fold sumsets","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO"],"primary_cat":"math.NT","authors_text":"Quan-Hui Yang, Yong-Gao Chen","submitted_at":"2014-06-28T02:33:20Z","abstract_excerpt":"Let $A=\\{a_0,a_1,\\ldots,a_{k-1}\\}$ be a set of $k$ integers. For any integer $h\\ge 1$ and any ordered $k$-tuple of positive integers $\\mathbf{r}=(r_0,r_1,\\ldots,r_{k-1})$, we define a general $h$-fold sumset, denoted by $h^{(\\mathbf{r})}A$, which is the set of all sums of $h$ elements of $A$, where $a_i$ appearing in the sum can be repeated at most $r_i$ times for $i=0,1,\\ldots,k-1$. In this paper, we give the best lower bound for $|h^{(\\mathbf{r})}A|$ in terms of $\\mathbf{r}$ and $h$ and determine the structure of the set $A$ when $|h^{(\\mathbf{r})}A|$ is minimal. This generalizes results of "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1406.7346","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"}