{"paper":{"title":"Additive structures imply more distances in $\\mathbb{F}_q^d$","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["math.CA","math.NT"],"primary_cat":"math.CO","authors_text":"Daewoong Cheong, Doowon Koh, Dung The Tran, Gennian Ge, Tao Zhang, Thang Pham","submitted_at":"2025-10-30T11:11:00Z","abstract_excerpt":"For a set $E \\subseteq \\mathbb{F}_q^d$, the distance set is defined as $\\Delta(E) := \\{\\|\\mathbf{x} - \\mathbf{y}\\| : \\mathbf{x}, \\mathbf{y} \\in E\\}$, where $\\|\\cdot\\|$ denotes the standard quadratic form. We investigate the Erd\\H{o}s--Falconer distance problem within the flexible class of $(u, s)$--Salem sets introduced by Jonathan M. Fraser, with emphasis on the even case $u = 4$. By exploiting the exact identity between $\\|\\widehat{E}\\|_4$ and the fourth additive energy $\\Lambda_4(E)$, we prove that quantitative gains in $\\Lambda_4(E)$ force the existence of many distances.\n  In particular, "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2510.26364","kind":"arxiv","version":3},"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/2510.26364/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"}