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In this note, we show that the number of distinct cubic distances determined by points in $A\\times A$ satisfies \\[|(A-A)^3+(A-A)^3|\\gg |A|^{8/7},\\] which improves a result due to Yazici, Murphy, Rudnev, and Shkredov. In addition, we investigate some new families of expanders in four and five variables.\n  We also give an explicit exponent of a problem of Bukh and Tsimerman, namely, we prove that \\[\\max \\left\\lbrace |A+A|, |f(A, A)|\\right\\rbrace\\gg |A|^{6/5},\\] where $f(x,"},"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":"1705.04255","kind":"arxiv","version":6},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2017-05-11T16:04:23Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"2fd690aa71f873048a0962e2096ef63286c2b5ec24e5453627810be177fccf50","abstract_canon_sha256":"09991b841c3e32fd1fe6ead3b12ffa674961f06e8ca2a523129b66b783928cba"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:12:00.457039Z","signature_b64":"ytIo9Ue0JFh9yXG6CI4Fz0acyEAjiV8Yu4pD+5yAe64zHjRPpPfFTimie4weu65kWSejBluqgHfTPDpauzQ7Bw==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"c5f14bc29dfd95ce039fb3a91e02848743c70fd2d12ed9b422faf7cb2ced0c6f","last_reissued_at":"2026-05-18T00:12:00.456049Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:12:00.456049Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Four-variable expanders over the prime fields","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.NT"],"primary_cat":"math.CO","authors_text":"Claudiu Valculescu, Doowon Koh, Hossein Nassajian Mojarrad, Thang Pham","submitted_at":"2017-05-11T16:04:23Z","abstract_excerpt":"Let $\\mathbb{F}_p$ be a prime field of order $p>2$, and $A$ be a set in $\\mathbb{F}_p$ with very small size in terms of $p$. 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