{"paper":{"title":"Multi-parameter projection theorems with applications to sums-products and finite point configurations in the Euclidean setting","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.CO","math.MG"],"primary_cat":"math.CA","authors_text":"A. Iosevich, B. Erdo\\u{g}an, D. Hart","submitted_at":"2011-06-28T00:27:29Z","abstract_excerpt":"In this paper we study multi-parameter projection theorems for fractal sets. With the help of these estimates, we recover results about the size of $A \\cdot A+...+A \\cdot A$, where $A$ is a subset of the real line of a given Hausdorff dimension, $A+A=\\{a+a': a,a' \\in A \\}$ and $A \\cdot A=\\{a \\cdot a': a,a' \\in A\\}$. We also use projection results and inductive arguments to show that if a Hausdorff dimension of a subset of ${\\Bbb R}^d$ is sufficiently large, then the ${k+1 \\choose 2}$-dimensional Lebesgue measure of the set of $k$-simplexes determined by this set is positive. The sharpness of t"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1106.5544","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"}