{"paper":{"title":"Next order energy asymptotics for Riesz potentials on flat tori","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"Brian Z. Simanek, Douglas P. Hardin, Edward B. Saff, Yujian Su","submitted_at":"2015-11-04T23:30:20Z","abstract_excerpt":"Let $\\Lambda$ be a lattice in ${\\bf R}^d$ with positive co-volume. Among $\\Lambda$-periodic $N$-point configurations, we consider the minimal renormalized Riesz $s$-energy $\\mathcal{E}_{s,\\Lambda}(N)$. While the dominant term in the asymptotic expansion of $\\mathcal{E}_{s,\\Lambda}(N)$ as $N$ goes to infinity in the long range case that $0<s<d$ (or $s=\\log$) can be obtained from classical potential theory, the next order term(s) require a different approach. Here we derive the form of the next order term or terms, namely for $s>0$ they are of the form $C_{s,d}|\\Lambda|^{-s/d}N^{1+s/d}$ and $-\\f"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1511.01552","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"}