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We show that large subsets of vector spaces over finite fields determine certain point configurations with prescribed distance structure. More specifically, we consider the complete graph with vertices as the points of $A \subseteq \mathbf{F}_q^d$ and edges assigned the algebraic distance between pairs of vertices. We prove nontrivial results on locating specified subgraphs of maximum vertex degree at most $t$ in dimensions $d \geq 2t$.
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참고문헌 (19)
참고문헌 신청
Michael Bennett / 2016 / Long paths in the distance graph over large subsets of vector spaces over finite fields / 대한수학회지 53(1):115~126
Michael Bennett / 2017 / Group actions and geometric combinatorics in Fdq / Forum Mathematicum 29(1)
Michael Bennett / 2016 / Finite chains inside thin subsets of ℝd / Analysis & PDE 9(3):597~614
Jeremy Chapman / 2012 / Pinned distance sets, k-simplices, Wolff’s exponent in finite fields and sum-product estimates / Mathematische Zeitschrift 271(1-2):63~93
N. Chatzikonstantinou / 2017 / Rigidity, graphs and Hausdorff dimension / arXiv preprint arXiv:1708.05919
Michael Bennett / 2017 / Group actions and geometric combinatorics in Fdq / Forum Mathematicum 29(1)
Michael Bennett / 2016 / Finite chains inside thin subsets of ℝd / Analysis & PDE 9(3):597~614
Jeremy Chapman / 2012 / Pinned distance sets, k-simplices, Wolff’s exponent in finite fields and sum-product estimates / Mathematische Zeitschrift 271(1-2):63~93
N. Chatzikonstantinou / 2017 / Rigidity, graphs and Hausdorff dimension / arXiv preprint arXiv:1708.05919
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