{"paper":{"title":"Lattice points close to families of surfaces, non-isotropic dilations and regularity of generalized Radon transforms","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.AP","math.CO","math.NT"],"primary_cat":"math.CA","authors_text":"Alex Iosevich, Krystal Taylor","submitted_at":"2011-03-09T00:15:17Z","abstract_excerpt":"We prove that if $\\phi: {\\Bbb R}^d \\times {\\Bbb R}^d \\to {\\Bbb R}$, $d \\ge 2$, is a homogeneous function, smooth away from the origin and having non-zero Monge-Ampere determinant away from the origin, then $$ R^{-d} # \\{(n,m) \\in {\\Bbb Z}^d \\times {\\Bbb Z}^d: |n|, |m| \\leq CR; R \\leq \\phi(n,m) \\leq R+\\delta \\} \\lesssim \\max \\{R^{d-2+\\frac{2}{d+1}}, R^{d-1} \\delta \\}.$$\n  This is a variable coefficient version of a result proved by Lettington in \\cite{L10}, extending a previous result by Andrews in \\cite{A63}, showing that if $B \\subset {\\Bbb R}^d$, $d \\ge 2$, is a symmetric convex body with a "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1103.1670","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":""},"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"}