{"paper":{"title":"Cross representations of additive complements of $r$-th powers","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Csaba S\\'andor, Yuchen Ding, Zihan Zhang","submitted_at":"2025-12-17T12:57:36Z","abstract_excerpt":"Let $\\mathbb{N}$ be the set of natural numbers and $\\mathcal{S}_r=\\big\\{1^r, 2^r, 3^r,\\cdots\\big\\}$ the set of $r$-th powers, where $r\\ge 2$ is a natural number. Let $\\mathcal{W}_r$ be an additive complement of $\\mathcal{S}_r$ and $$ f_r(n)=\\#\\big\\{(w,m^r)\\in \\mathcal{W}\\times \\mathcal{S}_r: n=w+m^r\\big\\}. $$ Motivated by a 1993 conjecture of Cilleruelo, we show that $$ \\sum_{n\\le N}f_r(n)-N\\gg_r N^{1-\\frac{1}{r}}. $$ Previously, the bound was only proved for $r=2$. In the case $r=2$, the lower bound above can be made more explicit as $$ \\sum_{n\\le N}f_2(n)-N\\gg N^{1/2}(\\log N)^{\\delta} $$ for"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2512.15407","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":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2512.15407/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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"}