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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"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"2512.15407","kind":"arxiv","version":5},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2025-12-17T12:57:36Z","cross_cats_sorted":[],"title_canon_sha256":"f89b2f86b074d99ecaec5872e84e124485b512a008741c447ec2ebbf199b1d80","abstract_canon_sha256":"427b6bc1c8ec525a4fe4137a350e67bbfad0f0dbbf437905d597907577651d9d"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-20T00:05:39.241217Z","signature_b64":"HG6PNqAAWiatUCv+/DeebIG+efiLNzP6kOga0JybIa7j9xiBiM36gkGcVl5U0i7KtGhfEQKe6xoevGZuIUZ6BQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"3ab38c298f9153e424d6dfec12e02b70e77ff7c092a9865df010dc42d0d45225","last_reissued_at":"2026-05-20T00:05:39.240294Z","signature_status":"signed_v1","first_computed_at":"2026-05-20T00:05:39.240294Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"2512.15407","created_at":"2026-05-20T00:05:39.240470+00:00"},{"alias_kind":"arxiv_version","alias_value":"2512.15407v5","created_at":"2026-05-20T00:05:39.240470+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2512.15407","created_at":"2026-05-20T00:05:39.240470+00:00"},{"alias_kind":"pith_short_12","alias_value":"HKZYYKMPSFJ6","created_at":"2026-05-20T00:05:39.240470+00:00"},{"alias_kind":"pith_short_16","alias_value":"HKZYYKMPSFJ6IJGW","created_at":"2026-05-20T00:05:39.240470+00:00"},{"alias_kind":"pith_short_8","alias_value":"HKZYYKMP","created_at":"2026-05-20T00:05:39.240470+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/HKZYYKMPSFJ6IJGW37WBFYBLOD","json":"https://pith.science/pith/HKZYYKMPSFJ6IJGW37WBFYBLOD.json","graph_json":"https://pith.science/api/pith-number/HKZYYKMPSFJ6IJGW37WBFYBLOD/graph.json","events_json":"https://pith.science/api/pith-number/HKZYYKMPSFJ6IJGW37WBFYBLOD/events.json","paper":"https://pith.science/paper/HKZYYKMP"},"agent_actions":{"view_html":"https://pith.science/pith/HKZYYKMPSFJ6IJGW37WBFYBLOD","download_json":"https://pith.science/pith/HKZYYKMPSFJ6IJGW37WBFYBLOD.json","view_paper":"https://pith.science/paper/HKZYYKMP","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2512.15407&json=true","fetch_graph":"https://pith.science/api/pith-number/HKZYYKMPSFJ6IJGW37WBFYBLOD/graph.json","fetch_events":"https://pith.science/api/pith-number/HKZYYKMPSFJ6IJGW37WBFYBLOD/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/HKZYYKMPSFJ6IJGW37WBFYBLOD/action/timestamp_anchor","attest_storage":"https://pith.science/pith/HKZYYKMPSFJ6IJGW37WBFYBLOD/action/storage_attestation","attest_author":"https://pith.science/pith/HKZYYKMPSFJ6IJGW37WBFYBLOD/action/author_attestation","sign_citation":"https://pith.science/pith/HKZYYKMPSFJ6IJGW37WBFYBLOD/action/citation_signature","submit_replication":"https://pith.science/pith/HKZYYKMPSFJ6IJGW37WBFYBLOD/action/replication_record"}},"created_at":"2026-05-20T00:05:39.240470+00:00","updated_at":"2026-05-20T00:05:39.240470+00:00"}